Theorem necon2bd 2458

Description: Contrapositive inference for inequality. (Contributed by NM, 13-Apr-2007.)

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 2401	. . 3 |- ( A =/= B <-> -. A = B )
3	1, 2	imbitrdi 161	. 2 |- ( ph -> ( ps -> -. A = B ) )
4	3	con2d 627	1 |- ( ph -> ( A = B -> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1395   =/= wne 2400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in1 617  ax-in2 618

This theorem depends on definitions:  df-bi 117  df-ne 2401

This theorem is referenced by:  disjiun  4081  map0g  6852  nneo  9573  zeo2  9576  bezoutr1  12594  coprm  12706  sqrt2irr  12724  dfphi2  12782  bj-charfunr  16341  nconstwlpolem  16605