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Mirrors > Home > MPE Home > Th. List > necon4bid | Structured version Visualization version GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.) |
Ref | Expression |
---|---|
necon4bid.1 | ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)) |
Ref | Expression |
---|---|
necon4bid | ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon4bid.1 | . . 3 ⊢ (𝜑 → (𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷)) | |
2 | 1 | necon2bbid 2984 | . 2 ⊢ (𝜑 → (𝐶 = 𝐷 ↔ ¬ 𝐴 ≠ 𝐵)) |
3 | nne 2944 | . 2 ⊢ (¬ 𝐴 ≠ 𝐵 ↔ 𝐴 = 𝐵) | |
4 | 2, 3 | bitr2di 291 | 1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 = wceq 1543 ≠ wne 2940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-ne 2941 |
This theorem is referenced by: nebi 3021 rpexp 16279 norm-i 29210 trlid0b 37929 |
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