Theorem pm2.61dne 2593

Description: Deduction eliminating an inequality in an antecedent. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypotheses

Ref	Expression
pm2.61dne.1	⊢ (φ → (A = B → ψ))
pm2.61dne.2	⊢ (φ → (A ≠ B → ψ))

Assertion

Ref	Expression
pm2.61dne	⊢ (φ → ψ)

Proof of Theorem pm2.61dne

Step	Hyp	Ref	Expression
1		pm2.61dne.2	. 2 ⊢ (φ → (A ≠ B → ψ))
2		nne 2520	. . 3 ⊢ (¬ A ≠ B ↔ A = B)
3		pm2.61dne.1	. . 3 ⊢ (φ → (A = B → ψ))
4	2, 3	syl5bi 208	. 2 ⊢ (φ → (¬ A ≠ B → ψ))
5	1, 4	pm2.61d 150	1 ⊢ (φ → ψ)

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:  pm2.61dane  2594