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Mirrors > Home > ILE Home > Th. List > necon3bii | GIF version |
Description: Inference from equality to inequality. (Contributed by NM, 23-Feb-2005.) |
Ref | Expression |
---|---|
necon3bii.1 | ⊢ (𝐴 = 𝐵 ↔ 𝐶 = 𝐷) |
Ref | Expression |
---|---|
necon3bii | ⊢ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon3bii.1 | . . 3 ⊢ (𝐴 = 𝐵 ↔ 𝐶 = 𝐷) | |
2 | 1 | necon3abii 2241 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐶 = 𝐷) |
3 | df-ne 2206 | . 2 ⊢ (𝐶 ≠ 𝐷 ↔ ¬ 𝐶 = 𝐷) | |
4 | 2, 3 | bitr4i 176 | 1 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ 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 |
This theorem depends on definitions: df-bi 110 df-ne 2206 |
This theorem is referenced by: necom 2289 negne0bi 7284 |
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