Theorem necon1ad 2584

Description: Contrapositive deduction for inequality. (Contributed by NM, 2-Apr-2007.)

Hypothesis

Ref	Expression
necon1ad.1	⊢ (φ → (¬ ψ → A = B))

Assertion

Ref	Expression
necon1ad	⊢ (φ → (A ≠ B → ψ))

Proof of Theorem necon1ad

Step	Hyp	Ref	Expression
1		df-ne 2519	. 2 ⊢ (A ≠ B ↔ ¬ A = B)
2		necon1ad.1	. . 3 ⊢ (φ → (¬ ψ → A = B))
3	2	con1d 116	. 2 ⊢ (φ → (¬ A = B → ψ))
4	1, 3	syl5bi 208	1 ⊢ (φ → (A ≠ B → ψ))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1642   ≠ wne 2517

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 2519

This theorem is referenced by:  nchoicelem15  6304