Theorem clel4 2848

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 2221	. . 3 |- ( x = B -> ( A e. x <-> A e. B ) )
3	1, 2	ceqsalv 2742	. 2 |- ( A. x ( x = B -> A e. x ) <-> A e. B )
4	3	bicomi 131	1 |- ( A e. B <-> A. x ( x = B -> A e. x ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1333   = wceq 1335   e. wcel 2128   _Vcvv 2712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714

This theorem is referenced by:  intpr  3839