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Mirrors > Home > NFE Home > Th. List > necon1ai | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 12-Feb-2007.) |
Ref | Expression |
---|---|
necon1ai.1 | ⊢ (¬ φ → A = B) |
Ref | Expression |
---|---|
necon1ai | ⊢ (A ≠ B → φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2518 | . 2 ⊢ (A ≠ B ↔ ¬ A = B) | |
2 | necon1ai.1 | . . 3 ⊢ (¬ φ → A = B) | |
3 | 2 | con1i 121 | . 2 ⊢ (¬ A = B → φ) |
4 | 1, 3 | sylbi 187 | 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: necon1i 2560 tz6.12i 5348 elfvdm 5351 elovex12 5648 |
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