Theorem necon3ai 1609

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon3ai.1	|- (ph -> A = B)

Assertion

Ref	Expression
necon3ai	|- (A =/= B -> -. ph)

Proof of Theorem necon3ai

Step	Hyp	Ref	Expression
1		df-ne 1590	. 2 |- (A =/= B <-> -. A = B)
2		necon3ai.1	. . 3 |- (ph -> A = B)
3	2	con3i 98	. 2 |- (-. A = B -> -. ph)
4	1, 3	sylbi 199	1 |- (A =/= B -> -. ph)

Syntax hints:  -. wn 2   -> wi 3   = wceq 958   =/= wne 1588

This theorem is referenced by:  disjsn2 2446  recextlem2 5695  lpbl 7877  oefil2 10552

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 1590

Copyright terms: Public domain