Theorem necon2bbii 1613

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon2bbii.1	|- (ph <-> A =/= B)

Assertion

Ref	Expression
necon2bbii	|- (A = B <-> -. ph)

Proof of Theorem necon2bbii

Step	Hyp	Ref	Expression
1		necon2bbii.1	. . . 4 |- (ph <-> A =/= B)
2	1	bicomi 172	. . 3 |- (A =/= B <-> ph)
3	2	necon1bbii 1609	. 2 |- (-. ph <-> A = B)
4	3	bicomi 172	1 |- (A = B <-> -. ph)

Syntax hints:  -. wn 2   <-> wb 146   = wceq 953   =/= wne 1577

This theorem is referenced by:  xpeq0 3453

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-ne 1579

Copyright terms: Public domain