Theorem neleq2 2475

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 2268	. . 3 |- ( A = B -> ( C e. A <-> C e. B ) )
2	1	notbid 668	. 2 |- ( A = B -> ( -. C e. A <-> -. C e. B ) )
3		df-nel 2471	. 2 |- ( C e/ A <-> -. C e. A )
4		df-nel 2471	. 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 1372   e. wcel 2175   e/ wnel 2470

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 615  ax-in2 616  ax-5 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-cleq 2197  df-clel 2200  df-nel 2471

This theorem is referenced by:  neleq12d  2476