Theorem clel2 3195

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 2523	. . 3 |- ( x = A -> ( x e. B <-> A e. B ) )
3	1, 2	ceqsalv 3098	. 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 1368   = wceq 1370   e. wcel 1758   _Vcvv 3070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3072

This theorem is referenced by:  snss  4099  mptelee  23278