Theorem necon3bd 2445

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 636	. 2 |- ( ph -> ( -. ps -> -. A = B ) )
3		df-ne 2403	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	imbitrrdi 162	1 |- ( ph -> ( -. ps -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1397   =/= wne 2402

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 619  ax-in2 620

This theorem depends on definitions:  df-bi 117  df-ne 2403

This theorem is referenced by:  nelne1  2492  nelne2  2493  nssne1  3285  nssne2  3286  disjne  3548  difsn  3810  nbrne1  4107  nbrne2  4108  ac6sfi  7087  indpi  7562  zneo  9581  pc2dvds  12905  pcadd  12915  oddprmdvds  12929  4sqlem11  12976  isnzr2  14201  lssvneln0  14390  lgsne0  15770  lgsquadlem2  15810  lgsquadlem3  15811