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Mirrors > Home > NFE Home > Th. List > necon4abid | GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.) |
Ref | Expression |
---|---|
necon4abid.1 | ⊢ (φ → (A ≠ B ↔ ¬ ψ)) |
Ref | Expression |
---|---|
necon4abid | ⊢ (φ → (A = B ↔ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2518 | . . 3 ⊢ (A ≠ B ↔ ¬ A = B) | |
2 | necon4abid.1 | . . 3 ⊢ (φ → (A ≠ B ↔ ¬ ψ)) | |
3 | 1, 2 | syl5bbr 250 | . 2 ⊢ (φ → (¬ A = B ↔ ¬ ψ)) |
4 | 3 | con4bid 284 | 1 ⊢ (φ → (A = B ↔ ψ)) |
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: necon4bbid 2581 |
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