Theorem neneqad 2325

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2267. One-way deduction form of df-ne 2247. (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 590	. 2 |- ( A = B -> -. ph )
3	2	necon2ai 2300	1 |- ( ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1285   =/= wne 2246

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578

This theorem depends on definitions:  df-bi 115  df-ne 2247

This theorem is referenced by:  ne0i  3264  nsuceq0g  4181  fidifsnen  6405  nqnq0pi  6690  xrlttri3  8948  expival  9575