Theorem necon3bd 2390

Description: Contrapositive law deduction for inequality. (Contributed by NM, 2-Apr-2007.) (Proof rewritten by Jim Kingdon, 15-May-2018.)

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 631	. 2 |- ( ph -> ( -. ps -> -. A = B ) )
3		df-ne 2348	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	imbitrrdi 162	1 |- ( ph -> ( -. ps -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   =/= wne 2347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-ne 2348

This theorem is referenced by:  nelne1  2437  nelne2  2438  nssne1  3215  nssne2  3216  disjne  3478  difsn  3731  nbrne1  4024  nbrne2  4025  ac6sfi  6900  indpi  7343  zneo  9356  pc2dvds  12331  pcadd  12341  oddprmdvds  12354  lssvneln0  13464  lgsne0  14478