Theorem clel4 2875

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 2241	. . 3 |- ( x = B -> ( A e. x <-> A e. B ) )
3	1, 2	ceqsalv 2769	. 2 |- ( A. x ( x = B -> A e. x ) <-> A e. B )
4	3	bicomi 132	1 |- ( A e. B <-> A. x ( x = B -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1351   = wceq 1353   e. wcel 2148   _Vcvv 2739

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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2741

This theorem is referenced by:  intpr  3878