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Mirrors > Home > NFE Home > Th. List > necon3bi | GIF version |
Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon3bi.1 | ⊢ (A = B → φ) |
Ref | Expression |
---|---|
necon3bi | ⊢ (¬ φ → A ≠ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nne 2520 | . . 3 ⊢ (¬ A ≠ B ↔ A = B) | |
2 | necon3bi.1 | . . 3 ⊢ (A = B → φ) | |
3 | 1, 2 | sylbi 187 | . 2 ⊢ (¬ A ≠ B → φ) |
4 | 3 | con1i 121 | 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-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: r19.2zb 3640 nchoicelem8 6296 |
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