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Mirrors > Home > NFE Home > Th. List > necon4bid | 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 2574 | . 2 ⊢ (φ → (C = D ↔ ¬ A ≠ B)) |
3 | nne 2520 | . 2 ⊢ (¬ A ≠ B ↔ A = B) | |
4 | 2, 3 | syl6rbb 253 | 1 ⊢ (φ → (A = B ↔ C = D)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 = wceq 1642 ≠ wne 2516 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 |
This theorem depends on definitions: df-bi 177 df-ne 2518 |
This theorem is referenced by: nebi 2587 |
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