Theorem neleq2 2500

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 2293	. . 3 |- ( A = B -> ( C e. A <-> C e. B ) )
2	1	notbid 671	. 2 |- ( A = B -> ( -. C e. A <-> -. C e. B ) )
3		df-nel 2496	. 2 |- ( C e/ A <-> -. C e. A )
4		df-nel 2496	. 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 1395   e. wcel 2200   e/ wnel 2495

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 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-clel 2225  df-nel 2496

This theorem is referenced by:  neleq12d  2501