New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > necon3ai | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 23-May-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon3ai.1 | ⊢ (φ → A = B) |
Ref | Expression |
---|---|
necon3ai | ⊢ (A ≠ B → ¬ φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon3ai.1 | . . 3 ⊢ (φ → A = B) | |
2 | nne 2520 | . . 3 ⊢ (¬ A ≠ B ↔ A = B) | |
3 | 1, 2 | sylibr 203 | . 2 ⊢ (φ → ¬ A ≠ B) |
4 | 3 | con2i 112 | 1 ⊢ (A ≠ B → ¬ φ) |
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: disjsn2 3787 fvunsn 5444 enadjlem1 6059 enadj 6060 |
Copyright terms: Public domain | W3C validator |