Theorem necon4bid 2583

Description: Contrapositive law deduction for inequality. (Contributed by NM, 29-Jun-2007.)

Hypothesis

Ref	Expression
necon4bid.1	⊢ (φ → (A ≠ B ↔ C ≠ D))

Assertion

Ref	Expression
necon4bid	⊢ (φ → (A = B ↔ C = D))

Proof of Theorem necon4bid

Step	Hyp	Ref	Expression
1		necon4bid.1	. . 3 ⊢ (φ → (A ≠ B ↔ C ≠ D))
2	1	necon2bbid 2575	. 2 ⊢ (φ → (C = D ↔ ¬ A ≠ B))
3		nne 2521	. 2 ⊢ (¬ A ≠ B ↔ A = B)
4	2, 3	syl6rbb 253	1 ⊢ (φ → (A = B ↔ C = D))

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:  nebi  2588