Theorem necon1bbii 2569

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

Hypothesis

Ref	Expression
necon1bbii.1	|- (A =/= B <-> ph)

Assertion

Ref	Expression
necon1bbii	|- (-. ph <-> A = B)

Proof of Theorem necon1bbii

Step	Hyp	Ref	Expression
1		df-ne 2519	. . 3 |- (A =/= B <-> -. A = B)
2		necon1bbii.1	. . 3 |- (A =/= B <-> ph)
3	1, 2	bitr3i 242	. 2 |- (-. A = B <-> ph)
4	3	con1bii 321	1 |- (-. ph <-> A = B)

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:  necon2bbii  2573  rabeq0  3573