Theorem necon2bbid 2575

Description: Contrapositive deduction for inequality. (Contributed by NM, 13-Apr-2007.)

Hypothesis

Ref	Expression
necon2bbid.1	⊢ (φ → (ψ ↔ A ≠ B))

Assertion

Ref	Expression
necon2bbid	⊢ (φ → (A = B ↔ ¬ ψ))

Proof of Theorem necon2bbid

Step	Hyp	Ref	Expression
1		necon2bbid.1	. . 3 ⊢ (φ → (ψ ↔ A ≠ B))
2		df-ne 2519	. . 3 ⊢ (A ≠ B ↔ ¬ A = B)
3	1, 2	syl6bb 252	. 2 ⊢ (φ → (ψ ↔ ¬ A = B))
4	3	con2bid 319	1 ⊢ (φ → (A = B ↔ ¬ ψ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ≠ wne 2517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-ne 2519

This theorem is used by:  necon4bid  2583