Theorem necon3ad 1605

Description: Contrapositive law deduction for inequality.

Hypothesis

Ref	Expression
necon3ad.1	|- (ph -> (ps -> A = B))

Assertion

Ref	Expression
necon3ad	|- (ph -> (A =/= B -> -. ps))

Proof of Theorem necon3ad

Step	Hyp	Ref	Expression
1		necon3ad.1	. . 3 |- (ph -> (ps -> A = B))
2	1	con3d 95	. 2 |- (ph -> (-. A = B -> -. ps))
3		df-ne 1590	. 2 |- (A =/= B <-> -. A = B)
4	2, 3	syl5ib 206	1 |- (ph -> (A =/= B -> -. ps))

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

This theorem is referenced by:  necon3d 1607  disjpss 2323  nlt1pi 5045  0nnei 7723  ocnelt 9165  hatomistic 10284  dmse1 10594

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

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