Theorem clel4 2909

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 2269	. . 3 |- ( x = B -> ( A e. x <-> A e. B ) )
3	1, 2	ceqsalv 2802	. 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 1371   = wceq 1373   e. wcel 2176   _Vcvv 2772

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774

This theorem is referenced by:  intpr  3917