Theorem necon1abii 2568

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.)

Hypothesis

Ref	Expression
necon1abii.1	⊢ (¬ φ ↔ A = B)

Assertion

Ref	Expression
necon1abii	⊢ (A ≠ B ↔ φ)

Proof of Theorem necon1abii

Step	Hyp	Ref	Expression
1		df-ne 2519	. 2 ⊢ (A ≠ B ↔ ¬ A = B)
2		necon1abii.1	. . 3 ⊢ (¬ φ ↔ A = B)
3	2	con1bii 321	. 2 ⊢ (¬ A = B ↔ φ)
4	1, 3	bitri 240	1 ⊢ (A ≠ B ↔ φ)

Syntax hints:  ¬ wn 3   ↔ 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:  necon2abii  2572