Theorem neneqad 2415

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2357. One-way deduction form of df-ne 2337. (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 617	. 2 |- ( A = B -> -. ph )
3	2	necon2ai 2390	1 |- ( ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1343   =/= wne 2336

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 604  ax-in2 605

This theorem depends on definitions:  df-bi 116  df-ne 2337

This theorem is referenced by:  ne0i  3415  nsuceq0g  4396  fidifsnen  6836  nqnq0pi  7379  xrlttri3  9733