Theorem necon3bd 2463

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 640	. 2 |- ( ph -> ( -. ps -> -. A = B ) )
3		df-ne 2421	. 2 |- ( A =/= B <-> -. A = B )
4	2, 3	imbitrrdi 162	1 |- ( ph -> ( -. ps -> A =/= B ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

This proof depends on definitions:  df-bi 117  df-ne 2421

This theorem is used by:  nelne1  2510  nelne2  2511  nssne1  3306  nssne2  3307  disjne  3578  difsn  3852  nbrne1  4149  nbrne2  4150  ac6sfi  7202  indpi  7709  zneo  9747  pc2dvds  13109  pcadd  13119  oddprmdvds  13133  4sqlem11  13180  isnzr2  14491  lssvneln0  14710  pellexlem1  16091  lgsne0  16157  lgsquadlem2  16197  lgsquadlem3  16198