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Mirrors > Home > NFE Home > Th. List > pm2.61ne | GIF version |
Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
pm2.61ne.1 | ⊢ (A = B → (ψ ↔ χ)) |
pm2.61ne.2 | ⊢ ((φ ∧ A ≠ B) → ψ) |
pm2.61ne.3 | ⊢ (φ → χ) |
Ref | Expression |
---|---|
pm2.61ne | ⊢ (φ → ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.61ne.2 | . . 3 ⊢ ((φ ∧ A ≠ B) → ψ) | |
2 | 1 | expcom 424 | . 2 ⊢ (A ≠ B → (φ → ψ)) |
3 | nne 2520 | . . 3 ⊢ (¬ A ≠ B ↔ A = B) | |
4 | pm2.61ne.3 | . . . 4 ⊢ (φ → χ) | |
5 | pm2.61ne.1 | . . . 4 ⊢ (A = B → (ψ ↔ χ)) | |
6 | 4, 5 | syl5ibr 212 | . . 3 ⊢ (A = B → (φ → ψ)) |
7 | 3, 6 | sylbi 187 | . 2 ⊢ (¬ A ≠ B → (φ → ψ)) |
8 | 2, 7 | pm2.61i 156 | 1 ⊢ (φ → ψ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 = 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-an 360 df-ne 2518 |
This theorem is referenced by: (None) |
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