Theorem pm2.21ddne 2503

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 2441	. 2 |- ( ph -> -. A = B )
4	1, 3	pm2.21dd 629	1 |- ( ph -> ps )

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

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-in2 624

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

This theorem is used by:  npnflt  10217  nmnfgt  10220  xlt2add  10282  xrbdtri  12042  divalglemex  12689  divalg  12691  znege1  12956  ennnfonelemex  13305  pw1ndom3  17020