Theorem neneqad 2499

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2441. One-way deduction form of df-ne 2421. (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 636	. 2 |- ( A = B -> -. ph )
3	2	necon2ai 2474	1 |- ( ph -> A =/= B )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   = wceq 1402   =/= wne 2420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624

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

This theorem is used by:  ne0i  3528  nsuceq0g  4563  fidifsnen  7172  nqnq0pi  7805  xrlttri3  10199  lcmval  12841  lcmcllem  12845