Theorem necon3bd 1601

Description: Contrapositive law deduction for inequality.

Hypothesis

Ref	Expression
necon3bd.1	|- (ph -> (A = B -> ps))

Assertion

Ref	Expression
necon3bd	|- (ph -> (-. ps -> A =/= B))

Proof of Theorem necon3bd

Step	Hyp	Ref	Expression
1		necon3bd.1	. . 3 |- (ph -> (A = B -> ps))
2	1	con3d 95	. 2 |- (ph -> (-. ps -> -. A = B))
3		df-ne 1585	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	syl6ibr 213	1 |- (ph -> (-. ps -> A =/= B))

Syntax hints:  -. wn 2   -> wi 3   = wceq 955   =/= wne 1583

This theorem is referenced by:  peano5 3149  inf3lem2 4597  zneo 6157

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 1585

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