Theorem pm2.21ddne 2878

Description: A contradiction implies anything. Equality/inequality deduction form. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
pm2.21ddne.1	|- ( ph -> A = B )
pm2.21ddne.2	|- ( ph -> A =/= B )

Assertion

Ref	Expression
pm2.21ddne	|- ( ph -> ps )

Proof of Theorem pm2.21ddne

Step	Hyp	Ref	Expression
1		pm2.21ddne.1	. 2 |- ( ph -> A = B )
2		pm2.21ddne.2	. . 3 |- ( ph -> A =/= B )
3	2	neneqd 2799	. 2 |- ( ph -> -. A = B )
4	1, 3	pm2.21dd 186	1 |- ( ph -> ps )

Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-ne 2795

This theorem is referenced by:  cshwshashlem2  15803  dprdsn  18435  coseq00topi  24254  tglndim0  25524  ncolncol  25541  footne  25615  sgnsub  30606  sgnmulsgn  30611  sgnmulsgp  30612  pconnconn  31213  osumcllem11N  35252  dochexmidlem8  36756  fnchoice  39188