Theorem clel2 3163

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

Hypothesis

Ref	Expression
clel2.1	|- A e. _V

Assertion

Ref	Expression
clel2	|- ( A e. B <-> A. x ( x = A -> x e. B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem clel2

Step	Hyp	Ref	Expression
1		clel2.1	. . 3 |- A e. _V
2		eleq1 2537	. . 3 |- ( x = A -> ( x e. B <-> A e. B ) )
3	1, 2	ceqsalv 3061	. 2 |- ( A. x ( x = A -> x e. B ) <-> A e. B )
4	3	bicomi 207	1 |- ( A e. B <-> A. x ( x = A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1450   = wceq 1452   e. wcel 1904   _Vcvv 3031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033

This theorem is referenced by:  snss  4087  mptelee  25004