Theorem necon2abid 2574

Description: Contrapositive deduction for inequality. (Contributed by NM, 18-Jul-2007.)

Hypothesis

Ref	Expression
necon2abid.1	⊢ (φ → (A = B ↔ ¬ ψ))

Assertion

Ref	Expression
necon2abid	⊢ (φ → (ψ ↔ A ≠ B))

Proof of Theorem necon2abid

Step	Hyp	Ref	Expression
1		necon2abid.1	. . 3 ⊢ (φ → (A = B ↔ ¬ ψ))
2	1	con2bid 319	. 2 ⊢ (φ → (ψ ↔ ¬ A = B))
3		df-ne 2519	. 2 ⊢ (A ≠ B ↔ ¬ A = B)
4	2, 3	syl6bbr 254	1 ⊢ (φ → (ψ ↔ A ≠ B))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = 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: (None)