New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > necon2bbid | GIF version |
Description: Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.) |
Ref | Expression |
---|---|
necon2bbid.1 | ⊢ (φ → (ψ ↔ A ≠ B)) |
Ref | Expression |
---|---|
necon2bbid | ⊢ (φ → (A = B ↔ ¬ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon2bbid.1 | . . 3 ⊢ (φ → (ψ ↔ A ≠ B)) | |
2 | df-ne 2518 | . . 3 ⊢ (A ≠ B ↔ ¬ A = B) | |
3 | 1, 2 | syl6bb 252 | . 2 ⊢ (φ → (ψ ↔ ¬ A = B)) |
4 | 3 | con2bid 319 | 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: necon4bid 2582 |
Copyright terms: Public domain | W3C validator |