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Mirrors > Home > ILE Home > Th. List > necon3bid | GIF version |
Description: Deduction from equality to inequality. (Contributed by NM, 23-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon3bid.1 | ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
Ref | Expression |
---|---|
necon3bid | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2246 | . 2 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
2 | necon3bid.1 | . . 3 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) | |
3 | 2 | necon3bbid 2285 | . 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 ↔ 𝐶 ≠ 𝐷)) |
4 | 1, 3 | syl5bb 190 | 1 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ 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 |
This theorem depends on definitions: df-bi 115 df-ne 2246 |
This theorem is referenced by: nebidc 2325 addneintrd 7296 addneintr2d 7297 negne0bd 7412 negned 7416 subne0d 7428 subne0ad 7430 subneintrd 7463 subneintr2d 7465 qapne 8724 xrlttri3 8872 sqne0 9541 cjne0 9795 absne0d 10073 sqrt2irraplemnn 10557 |
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