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Mirrors > Home > ILE Home > Th. List > neeq12i | GIF version |
Description: Inference for inequality. (Contributed by NM, 24-Jul-2012.) |
Ref | Expression |
---|---|
neeq1i.1 | ⊢ 𝐴 = 𝐵 |
neeq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
neeq12i | ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
2 | 1 | neeq2i 2221 | . 2 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐴 ≠ 𝐷) |
3 | neeq1i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
4 | 3 | neeq1i 2220 | . 2 ⊢ (𝐴 ≠ 𝐷 ↔ 𝐵 ≠ 𝐷) |
5 | 2, 4 | bitri 173 | 1 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐷) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 = wceq 1243 ≠ wne 2204 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-5 1336 ax-gen 1338 ax-4 1400 ax-17 1419 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 df-ne 2206 |
This theorem is referenced by: 3netr3g 2239 3netr4g 2240 |
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