Theorem neleq2 2502

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 2295	. . 3 |- ( A = B -> ( C e. A <-> C e. B ) )
2	1	notbid 673	. 2 |- ( A = B -> ( -. C e. A <-> -. C e. B ) )
3		df-nel 2498	. 2 |- ( C e/ A <-> -. C e. A )
4		df-nel 2498	. 2 |- ( C e/ B <-> -. C e. B )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( C e/ A <-> C e/ B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   e/ wnel 2497

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 619  ax-in2 620  ax-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227  df-nel 2498

This theorem is referenced by:  neleq12d  2503