Theorem neleq2 2344

Description: Equality theorem for negated membership. (Contributed by NM, 20-Nov-1994.)

Assertion

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

Proof of Theorem neleq2

Step	Hyp	Ref	Expression
1		eleq2 2142	. . 3 |- ( A = B -> ( C e. A <-> C e. B ) )
2	1	notbid 624	. 2 |- ( A = B -> ( -. C e. A <-> -. C e. B ) )
3		df-nel 2340	. 2 |- ( C e/ A <-> -. C e. A )
4		df-nel 2340	. 2 |- ( C e/ B <-> -. C e. B )
5	2, 3, 4	3bitr4g 221	1 |- ( A = B -> ( C e/ A <-> C e/ B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   e/ wnel 2339

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 576  ax-in2 577  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-cleq 2074  df-clel 2077  df-nel 2340

This theorem is referenced by:  neleq12d  2345