Theorem necon1abii 2693

Description: Contrapositive inference for inequality. (Contributed by NM, 17-Mar-2007.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Hypothesis

Ref	Expression
necon1abii.1	|- ( -. ph <-> A = B )

Assertion

Ref	Expression
necon1abii	|- ( A =/= B <-> ph )

Proof of Theorem necon1abii

Step	Hyp	Ref	Expression
1		notnot 292	. 2 |- ( ph <-> -. -. ph )
2		necon1abii.1	. . 3 |- ( -. ph <-> A = B )
3	2	necon3bbii 2692	. 2 |- ( -. -. ph <-> A =/= B )
4	1, 3	bitr2i 253	1 |- ( A =/= B <-> ph )

Syntax hints:   -. wn 3   <-> wb 187   = wceq 1437   =/= wne 2625

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 188  df-ne 2627

This theorem is referenced by:  necon2abii  2697  marypha1lem  7953  npomex  9420  uniinn0  28002