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Mirrors > Home > NFE Home > Th. List > necon3bd | GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon3bd.1 | ⊢ (φ → (A = B → ψ)) |
Ref | Expression |
---|---|
necon3bd | ⊢ (φ → (¬ ψ → A ≠ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nne 2520 | . . 3 ⊢ (¬ A ≠ B ↔ A = B) | |
2 | necon3bd.1 | . . 3 ⊢ (φ → (A = B → ψ)) | |
3 | 1, 2 | syl5bi 208 | . 2 ⊢ (φ → (¬ A ≠ B → ψ)) |
4 | 3 | con1d 116 | 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-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: nelne1 2605 nelne2 2606 nssne1 3327 nssne2 3328 disjne 3596 difsn 3845 nbrne1 4656 nbrne2 4657 |
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