Theorem necon2d 2567

Description: Contrapositive inference for inequality. (Contributed by NM, 28-Dec-2008.)

Hypothesis

Ref	Expression
necon2d.1	⊢ (φ → (A = B → C ≠ D))

Assertion

Ref	Expression
necon2d	⊢ (φ → (C = D → A ≠ B))

Proof of Theorem necon2d

Step	Hyp	Ref	Expression
1		necon2d.1	. . 3 ⊢ (φ → (A = B → C ≠ D))
2		df-ne 2519	. . 3 ⊢ (C ≠ D ↔ ¬ C = D)
3	1, 2	syl6ib 217	. 2 ⊢ (φ → (A = B → ¬ C = D))
4	3	necon2ad 2565	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: (None)