Theorem neleq1 2475

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 2268	. . 3 |- ( A = B -> ( A e. C <-> B e. C ) )
2	1	notbid 669	. 2 |- ( A = B -> ( -. A e. C <-> -. B e. C ) )
3		df-nel 2472	. 2 |- ( A e/ C <-> -. A e. C )
4		df-nel 2472	. 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 1373   e. wcel 2176   e/ wnel 2471

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-cleq 2198  df-clel 2201  df-nel 2472

This theorem is referenced by:  neleq12d  2477  ruALT  4599