Theorem necon2bd 1607

Description: Contrapositive inference for inequality.

Hypothesis

Ref	Expression
necon2bd.1	|- (ph -> (ps -> A =/= B))

Assertion

Ref	Expression
necon2bd	|- (ph -> (A = B -> -. ps))

Proof of Theorem necon2bd

Step	Hyp	Ref	Expression
1		necon2bd.1	. . 3 |- (ph -> (ps -> A =/= B))
2		df-ne 1579	. . 3 |- (A =/= B <-> -. A = B)
3	1, 2	syl6ib 212	. 2 |- (ph -> (ps -> -. A = B))
4	3	con2d 91	1 |- (ph -> (A = B -> -. ps))

Syntax hints:  -. wn 2   -> wi 3   = wceq 953   =/= wne 1577

This theorem is referenced by:  fr2nr 2915  fr3nr 2916  stadd 10083

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

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