Theorem necon4d 2580

Description: Contrapositive inference for inequality. (Contributed by NM, 2-Apr-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon4d.1	⊢ (φ → (A ≠ B → C ≠ D))

Assertion

Ref	Expression
necon4d	⊢ (φ → (C = D → A = B))

Proof of Theorem necon4d

Step	Hyp	Ref	Expression
1		necon4d.1	. . 3 ⊢ (φ → (A ≠ B → C ≠ D))
2	1	necon2bd 2566	. 2 ⊢ (φ → (C = D → ¬ A ≠ B))
3		nne 2521	. 2 ⊢ (¬ A ≠ B ↔ A = B)
4	2, 3	syl6ib 217	1 ⊢ (φ → (C = D → A = B))

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:  tfin11  4494