Theorem necon4abid 2580

Description: Contrapositive law deduction for inequality. (Contributed by NM, 11-Jan-2008.)

Hypothesis

Ref	Expression
necon4abid.1	⊢ (φ → (A ≠ B ↔ ¬ ψ))

Assertion

Ref	Expression
necon4abid	⊢ (φ → (A = B ↔ ψ))

Proof of Theorem necon4abid

Step	Hyp	Ref	Expression
1		df-ne 2518	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
2		necon4abid.1	. . 3 ⊢ (φ → (A ≠ B ↔ ¬ ψ))
3	1, 2	syl5bbr 250	. 2 ⊢ (φ → (¬ A = B ↔ ¬ ψ))
4	3	con4bid 284	1 ⊢ (φ → (A = B ↔ ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = 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:  necon4bbid  2581