Theorem neleq2 2903

Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.) (Proof shortened by Wolf Lammen, 25-Nov-2019.)

Assertion

Ref	Expression
neleq2	|- ( A = B -> ( C e/ A <-> C e/ B ) )

Proof of Theorem neleq2

Step	Hyp	Ref	Expression
1		eqidd 2623	. 2 |- ( A = B -> C = C )
2		id 22	. 2 |- ( A = B -> A = B )
3	1, 2	neleq12d 2901	1 |- ( A = B -> ( C e/ A <-> C e/ B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e/ wnel 2897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-nel 2898

This theorem is referenced by:  noinfep  8557  wrdlndm  13321  isfbas  21633  upgrreslem  26196  umgrreslem  26197  nbgrnvtx0  26237  nbupgrres  26266  eupth2lem3lem6  27093  frgrncvvdeqlem1  27163  frgrwopreglem4a  27174