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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 2258 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 = wceq 1284 ≠ wne 2245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-4 1440 ax-17 1459 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-ne 2246 |
This theorem is referenced by: neeq12i 2262 eqnetri 2268 syl5eqner 2276 rabn0r 3271 |
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