Theorem neleq1 2446

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

Assertion

Ref	Expression
neleq1	|- ( A = B -> ( A e/ C <-> B e/ C ) )

Proof of Theorem neleq1

Step	Hyp	Ref	Expression
1		eleq1 2240	. . 3 |- ( A = B -> ( A e. C <-> B e. C ) )
2	1	notbid 667	. 2 |- ( A = B -> ( -. A e. C <-> -. B e. C ) )
3		df-nel 2443	. 2 |- ( A e/ C <-> -. A e. C )
4		df-nel 2443	. 2 |- ( B e/ C <-> -. B e. C )
5	2, 3, 4	3bitr4g 223	1 |- ( A = B -> ( A e/ C <-> B e/ C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2148   e/ wnel 2442

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 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-clel 2173  df-nel 2443

This theorem is referenced by:  neleq12d  2448  ruALT  4552