diff --git a/cranelift/codegen/src/isle_prelude.rs b/cranelift/codegen/src/isle_prelude.rs
index 23270f0c49e3..58f42f48680a 100644
--- a/cranelift/codegen/src/isle_prelude.rs
+++ b/cranelift/codegen/src/isle_prelude.rs
@@ -509,6 +509,16 @@ macro_rules! isle_common_prelude_methods {
             Imm64::new(x as i64)
         }
 
+        #[inline]
+        fn imm64_masked(&mut self, ty: Type, x: u64) -> Imm64 {
+            debug_assert!(ty.bits() <= 64);
+            // Careful: we can't do `(1 << bits) - 1` because that
+            // would overflow for `bits == 64`. Instead,
+            // right-shift an all-ones mask.
+            let mask = u64::MAX >> (64 - ty.bits());
+            Imm64::new((x & mask) as i64)
+        }
+
         #[inline]
         fn simm32(&mut self, x: Imm64) -> Option<u32> {
             let x64: i64 = x.into();
diff --git a/cranelift/codegen/src/opts/algebraic.isle b/cranelift/codegen/src/opts/algebraic.isle
index e4638222fe21..d407474131ef 100644
--- a/cranelift/codegen/src/opts/algebraic.isle
+++ b/cranelift/codegen/src/opts/algebraic.isle
@@ -207,3 +207,36 @@
 (rule (simplify
        (select ty (uextend _ c @ (icmp _ _ _ _)) x y))
       (select ty c x y))
+
+;; `x == x` is always true for integers; `x != x` is false. Strict
+;; inequalities are false, and loose inequalities are true.
+(rule (simplify
+       (icmp ty (IntCC.Equal) x x))
+      (iconst ty (imm64 1)))
+(rule (simplify
+       (icmp ty (IntCC.NotEqual) x x))
+      (iconst ty (imm64 0)))
+(rule (simplify
+       (icmp ty (IntCC.UnsignedGreaterThan) x x))
+      (iconst ty (imm64 0)))
+(rule (simplify
+       (icmp ty (IntCC.UnsignedGreaterThanOrEqual) x x))
+      (iconst ty (imm64 1)))
+(rule (simplify
+       (icmp ty (IntCC.SignedGreaterThan) x x))
+      (iconst ty (imm64 0)))
+(rule (simplify
+       (icmp ty (IntCC.SignedGreaterThanOrEqual) x x))
+      (iconst ty (imm64 1)))
+(rule (simplify
+       (icmp ty (IntCC.UnsignedLessThan) x x))
+      (iconst ty (imm64 0)))
+(rule (simplify
+       (icmp ty (IntCC.UnsignedLessThanOrEqual) x x))
+      (iconst ty (imm64 1)))
+(rule (simplify
+       (icmp ty (IntCC.SignedLessThan) x x))
+      (iconst ty (imm64 0)))
+(rule (simplify
+       (icmp ty (IntCC.SignedLessThanOrEqual) x x))
+      (iconst ty (imm64 1)))
diff --git a/cranelift/codegen/src/opts/cprop.isle b/cranelift/codegen/src/opts/cprop.isle
index ac6ce778e006..d7ba06dcf634 100644
--- a/cranelift/codegen/src/opts/cprop.isle
+++ b/cranelift/codegen/src/opts/cprop.isle
@@ -4,56 +4,56 @@
        (iadd (fits_in_64 ty)
              (iconst ty (u64_from_imm64 k1))
              (iconst ty (u64_from_imm64 k2))))
-      (subsume (iconst ty (imm64 (u64_add k1 k2)))))
+      (subsume (iconst ty (imm64_masked ty (u64_add k1 k2)))))
 
 (rule (simplify
        (isub (fits_in_64 ty)
              (iconst ty (u64_from_imm64 k1))
              (iconst ty (u64_from_imm64 k2))))
-      (subsume (iconst ty (imm64 (u64_sub k1 k2)))))
+      (subsume (iconst ty (imm64_masked ty (u64_sub k1 k2)))))
 
 (rule (simplify
        (imul (fits_in_64 ty)
              (iconst ty (u64_from_imm64 k1))
              (iconst ty (u64_from_imm64 k2))))
-      (subsume (iconst ty (imm64 (u64_mul k1 k2)))))
+      (subsume (iconst ty (imm64_masked ty (u64_mul k1 k2)))))
 
 (rule (simplify
        (sdiv (fits_in_64 ty)
              (iconst ty (u64_from_imm64 k1))
              (iconst ty (u64_from_imm64 k2))))
       (if-let d (u64_sdiv k1 k2))
-      (subsume (iconst ty (imm64 d))))
+      (subsume (iconst ty (imm64_masked ty d))))
 
 (rule (simplify
        (udiv (fits_in_64 ty)
              (iconst ty (u64_from_imm64 k1))
              (iconst ty (u64_from_imm64 k2))))
       (if-let d (u64_udiv k1 k2))
-      (subsume (iconst ty (imm64 d))))
+      (subsume (iconst ty (imm64_masked ty d))))
 
 (rule (simplify
        (bor (fits_in_64 ty)
             (iconst ty (u64_from_imm64 k1))
             (iconst ty (u64_from_imm64 k2))))
-      (subsume (iconst ty (imm64 (u64_or k1 k2)))))
+      (subsume (iconst ty (imm64_masked ty (u64_or k1 k2)))))
 
 (rule (simplify
        (band (fits_in_64 ty)
             (iconst ty (u64_from_imm64 k1))
             (iconst ty (u64_from_imm64 k2))))
-      (subsume (iconst ty (imm64 (u64_and k1 k2)))))
+      (subsume (iconst ty (imm64_masked ty (u64_and k1 k2)))))
 
 (rule (simplify
        (bxor (fits_in_64 ty)
             (iconst ty (u64_from_imm64 k1))
             (iconst ty (u64_from_imm64 k2))))
-      (subsume (iconst ty (imm64 (u64_xor k1 k2)))))
+      (subsume (iconst ty (imm64_masked ty (u64_xor k1 k2)))))
 
 (rule (simplify
        (bnot (fits_in_64 ty)
             (iconst ty (u64_from_imm64 k))))
-      (subsume (iconst ty (imm64 (u64_not k)))))
+      (subsume (iconst ty (imm64_masked ty (u64_not k)))))
 
 ;; Canonicalize via commutativity: push immediates to the right.
 ;;
@@ -99,23 +99,23 @@
 (rule (simplify (isub ty
                       (isub ty x (iconst ty (u64_from_imm64 k1)))
                       (iconst ty (u64_from_imm64 k2))))
-      (isub ty x (iconst ty (imm64 (u64_add k1 k2)))))
+      (isub ty x (iconst ty (imm64_masked ty (u64_add k1 k2)))))
 (rule (simplify (isub ty
                       (isub ty (iconst ty (u64_from_imm64 k1)) x)
                       (iconst ty (u64_from_imm64 k2))))
-      (isub ty (iconst ty (imm64 (u64_sub k1 k2))) x))
+      (isub ty (iconst ty (imm64_masked ty (u64_sub k1 k2))) x))
 (rule (simplify (isub ty
                       (iadd ty x (iconst ty (u64_from_imm64 k1)))
                       (iconst ty (u64_from_imm64 k2))))
-      (isub ty x (iconst ty (imm64 (u64_sub k2 k1)))))
+      (isub ty x (iconst ty (imm64_masked ty (u64_sub k2 k1)))))
 (rule (simplify (iadd ty
                       (isub ty x (iconst ty (u64_from_imm64 k1)))
                       (iconst ty (u64_from_imm64 k2))))
-      (iadd ty x (iconst ty (imm64 (u64_sub k2 k1)))))
+      (iadd ty x (iconst ty (imm64_masked ty (u64_sub k2 k1)))))
 (rule (simplify (iadd ty
                       (isub ty (iconst ty (u64_from_imm64 k1)) x)
                       (iconst ty (u64_from_imm64 k2))))
-      (isub ty (iconst ty (imm64 (u64_add k1 k2))) x))
+      (isub ty (iconst ty (imm64_masked ty (u64_add k1 k2))) x))
 
 (rule (simplify
        (imul ty (imul ty x k1 @ (iconst ty _)) k2 @ (iconst ty _)))
@@ -138,4 +138,3 @@
       y)
 
 ;; TODO: fadd, fsub, fmul, fdiv, fneg, fabs
-
diff --git a/cranelift/codegen/src/prelude.isle b/cranelift/codegen/src/prelude.isle
index 0f17b6b442cc..5a0aa3376cbf 100644
--- a/cranelift/codegen/src/prelude.isle
+++ b/cranelift/codegen/src/prelude.isle
@@ -349,6 +349,10 @@
 (decl pure imm64 (u64) Imm64)
 (extern constructor imm64 imm64)
 
+;; Create a new Imm64, masked to the width of the given type.
+(decl pure imm64_masked (Type u64) Imm64)
+(extern constructor imm64_masked imm64_masked)
+
 ;; Extract a `u64` from an `Ieee32`.
 (decl u64_from_ieee32 (u64) Ieee32)
 (extern extractor infallible u64_from_ieee32 u64_from_ieee32)
diff --git a/cranelift/filetests/filetests/egraph/cprop.clif b/cranelift/filetests/filetests/egraph/cprop.clif
new file mode 100644
index 000000000000..bc76e1d546e7
--- /dev/null
+++ b/cranelift/filetests/filetests/egraph/cprop.clif
@@ -0,0 +1,43 @@
+test optimize
+set opt_level=none
+set use_egraphs=true
+target x86_64
+
+function %f0() -> i8 {
+block0:
+    v1 = iconst.i8 51
+    v2 = imul.i8 v1, v1
+    return v2
+}
+
+; check: v9 = iconst.i8 41
+; nextln: return v9
+
+function %f1() -> i16 {
+block0:
+    v1 = iconst.i16 1
+    v2 = bnot.i16 v1
+    return v2
+}
+
+; check: v3 = iconst.i16 0xfffe
+; nextln: return v3
+
+function %f2(i8) -> i8 {
+block0(v1: i8):
+    v2 = icmp eq v1, v1
+    return v2
+}
+
+; check: v3 = iconst.i8 1
+; check: return v3
+
+function %f3(i8) -> i8 {
+block0(v1: i8):
+    v2 = icmp ne v1, v1
+    return v2
+}
+
+; check: v3 = iconst.i8 0
+; check: return v3
+
diff --git a/cranelift/filetests/filetests/egraph/issue-5405.clif b/cranelift/filetests/filetests/egraph/issue-5405.clif
new file mode 100644
index 000000000000..c453875de5cf
--- /dev/null
+++ b/cranelift/filetests/filetests/egraph/issue-5405.clif
@@ -0,0 +1,16 @@
+test interpret
+test run
+set opt_level=speed_and_size
+set use_egraphs=true
+target aarch64
+
+function %a(i64) -> i8 system_v {
+block0(v0: i64):
+    v6 = iconst.i8 51
+    v17 = imul v6, v6  ; v6 = 51, v6 = 51
+    v18 = icmp eq v17, v17
+    v52 = imul v18, v18
+    return v52
+}
+
+; run: %a(129) == 1