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Mirrors > Home > ILE Home > Th. List > neeq1i | GIF version |
Description: Inference for inequality. (Contributed by NM, 29-Apr-2005.) |
Ref | Expression |
---|---|
neeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
neeq1i | ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | neeq1 2322 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1332 ≠ wne 2309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-5 1424 ax-gen 1426 ax-4 1488 ax-17 1507 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-cleq 2133 df-ne 2310 |
This theorem is referenced by: neeq12i 2326 eqnetri 2332 eqnetrrid 2340 rabn0r 3394 |
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