Theorem neleq2 2302

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 2101	. . 3 |- ( A = B -> ( C e. A <-> C e. B ) )
2	1	notbid 592	. 2 |- ( A = B -> ( -. C e. A <-> -. C e. B ) )
3		df-nel 2207	. 2 |- ( C e/ A <-> -. C e. A )
4		df-nel 2207	. 2 |- ( C e/ B <-> -. C e. B )
5	2, 3, 4	3bitr4g 212	1 |- ( A = B -> ( C e/ A <-> C e/ B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   e/ wnel 2205

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-cleq 2033  df-clel 2036  df-nel 2207

This theorem is referenced by:  neleq12d  2303