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Mirrors > Home > NFE Home > Th. List > necon1bi | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 18-Mar-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon1bi.1 | ⊢ (A ≠ B → φ) |
Ref | Expression |
---|---|
necon1bi | ⊢ (¬ φ → A = B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon1bi.1 | . . 3 ⊢ (A ≠ B → φ) | |
2 | 1 | con3i 127 | . 2 ⊢ (¬ φ → ¬ A ≠ B) |
3 | nne 2520 | . 2 ⊢ (¬ A ≠ B ↔ A = B) | |
4 | 2, 3 | sylib 188 | 1 ⊢ (¬ φ → 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: mapprc 6004 |
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