Theorem necon2adOLD 2056

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon2ad.1	|- (ph -> (A = B -> -. ps))

Assertion

Ref	Expression
necon2adOLD	|- (ph -> (ps -> A =/= B))

Proof of Theorem necon2adOLD

Step	Hyp	Ref	Expression
1		necon2ad.1	. . 3 |- (ph -> (A = B -> -. ps))
2	1	con2d 107	. 2 |- (ph -> (ps -> -. A = B))
3		df-ne 2019	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	syl6ibr 230	1 |- (ph -> (ps -> A =/= B))

Syntax hints:  -. wn 2   -> wi 3   = wceq 1298   =/= wne 2017

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 164  df-ne 2019

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