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	syl6ibr 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  3213  nssne2  3214  disjne  3476  difsn  3729  nbrne1  4020  nbrne2  4021  ac6sfi  6893  indpi  7336  zneo  9348  pc2dvds  12319  pcadd  12329  oddprmdvds  12342  lgsne0  14221