Theorem necon3bd 2410

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 632	. 2 |- ( ph -> ( -. ps -> -. A = B ) )
3		df-ne 2368	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	imbitrrdi 162	1 |- ( ph -> ( -. ps -> A =/= B ) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   =/= wne 2367

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 615  ax-in2 616

This theorem depends on definitions:  df-bi 117  df-ne 2368

This theorem is referenced by:  nelne1  2457  nelne2  2458  nssne1  3242  nssne2  3243  disjne  3505  difsn  3760  nbrne1  4053  nbrne2  4054  ac6sfi  6968  indpi  7428  zneo  9446  pc2dvds  12526  pcadd  12536  oddprmdvds  12550  4sqlem11  12597  isnzr2  13818  lssvneln0  14007  lgsne0  15387  lgsquadlem2  15427  lgsquadlem3  15428