Theorem clel4 2731

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
clel4.1	|- B e. _V

Assertion

Ref	Expression
clel4	|- ( A e. B <-> A. x ( x = B -> A e. x ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem clel4

Step	Hyp	Ref	Expression
1		clel4.1	. . 3 |- B e. _V
2		eleq2 2142	. . 3 |- ( x = B -> ( A e. x <-> A e. B ) )
3	1, 2	ceqsalv 2629	. 2 |- ( A. x ( x = B -> A e. x ) <-> A e. B )
4	3	bicomi 130	1 |- ( A e. B <-> A. x ( x = B -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   e. wcel 1433   _Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  intpr  3668