Theorem necon3bi 2558

Description: Contrapositive inference for inequality. (Contributed by NM, 1-Jun-2007.) (Proof shortened by Andrew Salmon, 25-May-2011.)

Hypothesis

Ref	Expression
necon3bi.1	⊢ (A = B → φ)

Assertion

Ref	Expression
necon3bi	⊢ (¬ φ → A ≠ B)

Proof of Theorem necon3bi

Step	Hyp	Ref	Expression
1		nne 2521	. . 3 ⊢ (¬ A ≠ B ↔ A = B)
2		necon3bi.1	. . 3 ⊢ (A = B → φ)
3	1, 2	sylbi 187	. 2 ⊢ (¬ A ≠ B → φ)
4	3	con1i 121	1 ⊢ (¬ φ → 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:  r19.2zb  3641  nchoicelem8  6297