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Mirrors > Home > NFE Home > Th. List > necon2ad | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 19-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon2ad.1 | ⊢ (φ → (A = B → ¬ ψ)) |
Ref | Expression |
---|---|
necon2ad | ⊢ (φ → (ψ → A ≠ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nne 2520 | . . 3 ⊢ (¬ A ≠ B ↔ A = B) | |
2 | necon2ad.1 | . . 3 ⊢ (φ → (A = B → ¬ ψ)) | |
3 | 1, 2 | syl5bi 208 | . 2 ⊢ (φ → (¬ A ≠ B → ¬ ψ)) |
4 | 3 | con4d 97 | 1 ⊢ (φ → (ψ → A ≠ B)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1642 ≠ wne 2516 |
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 177 df-ne 2518 |
This theorem is referenced by: necon2d 2566 nulnnn 4556 enadjlem1 6059 |
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