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Mirrors > Home > NFE Home > Th. List > necon1d | GIF version |
Description: Contrapositive law deduction for inequality. (Contributed by NM, 28-Dec-2008.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
necon1d.1 | ⊢ (φ → (A ≠ B → C = D)) |
Ref | Expression |
---|---|
necon1d | ⊢ (φ → (C ≠ D → A = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | necon1d.1 | . . 3 ⊢ (φ → (A ≠ B → C = D)) | |
2 | nne 2520 | . . 3 ⊢ (¬ C ≠ D ↔ C = D) | |
3 | 1, 2 | syl6ibr 218 | . 2 ⊢ (φ → (A ≠ B → ¬ C ≠ D)) |
4 | 3 | necon4ad 2577 | 1 ⊢ (φ → (C ≠ D → 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: (None) |
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