Theorem necon1abid 1618

Description: Contrapositive deduction for inequality.

Hypothesis

Ref	Expression
necon1abid.1	|- (ph -> (-. ps <-> A = B))

Assertion

Ref	Expression
necon1abid	|- (ph -> (A =/= B <-> ps))

Proof of Theorem necon1abid

Step	Hyp	Ref	Expression
1		necon1abid.1	. . 3 |- (ph -> (-. ps <-> A = B))
2	1	con1bid 527	. 2 |- (ph -> (-. A = B <-> ps))
3		df-ne 1587	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	syl5bb 532	1 |- (ph -> (A =/= B <-> ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   = wceq 956   =/= wne 1585

This theorem is referenced by:  lttri2t 5513  xrlttri2t 5555  ioon0t 6369

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-an 225  df-ne 1587

Copyright terms: Public domain