Theorem neleq1 2405

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 2200	. . 3 |- ( A = B -> ( A e. C <-> B e. C ) )
2	1	notbid 656	. 2 |- ( A = B -> ( -. A e. C <-> -. B e. C ) )
3		df-nel 2402	. 2 |- ( A e/ C <-> -. A e. C )
4		df-nel 2402	. 2 |- ( B e/ C <-> -. B e. C )
5	2, 3, 4	3bitr4g 222	1 |- ( A = B -> ( A e/ C <-> B e/ C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   = wceq 1331   e. wcel 1480   e/ wnel 2401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133  df-nel 2402

This theorem is referenced by:  neleq12d  2407  ruALT  4461