Theorem clel2 3236

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 2529	. . 3 |- ( x = A -> ( x e. B <-> A e. B ) )
3	1, 2	ceqsalv 3137	. 2 |- ( A. x ( x = A -> x e. B ) <-> A e. B )
4	3	bicomi 202	1 |- ( A e. B <-> A. x ( x = A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 184   A.wal 1393   = wceq 1395   e. wcel 1819   _Vcvv 3109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111

This theorem is referenced by:  snss  4156  mptelee  24324