Theorem neneqad 2387

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2329. One-way deduction form of df-ne 2309. (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 616	. 2 |- ( A = B -> -. ph )
3	2	necon2ai 2362	1 |- ( ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1331   =/= wne 2308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604

This theorem depends on definitions:  df-bi 116  df-ne 2309

This theorem is referenced by:  ne0i  3369  nsuceq0g  4340  fidifsnen  6764  nqnq0pi  7246  xrlttri3  9583