Theorem neneqad 2426

Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2368. One-way deduction form of df-ne 2348. (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 627	. 2 |- ( A = B -> -. ph )
3	2	necon2ai 2401	1 |- ( ph -> A =/= B )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   =/= wne 2347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615

This theorem depends on definitions:  df-bi 117  df-ne 2348

This theorem is referenced by:  ne0i  3429  nsuceq0g  4417  fidifsnen  6867  nqnq0pi  7434  xrlttri3  9793