Theorem neneqad 2586

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2532. One-way deduction form of df-ne 2518. (Contributed by David Moews, 28-Feb-2017.)

Hypothesis

Ref	Expression
neneqad.1	|- (ph -> -. A = B)

Assertion

Ref	Expression
neneqad	|- (ph -> A =/= B)

Proof of Theorem neneqad

Step	Hyp	Ref	Expression
1		neneqad.1	. . 3 |- (ph -> -. A = B)
2	1	con2i 112	. 2 |- (A = B -> -. ph)
3	2	necon2ai 2561	1 |- (ph -> A =/= B)

Syntax hints:  -. wn 3   -> wi 4   = wceq 1642   =/= wne 2516

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 177  df-ne 2518

This theorem is referenced by: (None)