Theorem clel2 3339

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 2689	. . 3 |- ( x = A -> ( x e. B <-> A e. B ) )
3	1, 2	ceqsalv 3233	. 2 |- ( A. x ( x = A -> x e. B ) <-> A e. B )
4	3	bicomi 214	1 |- ( A e. B <-> A. x ( x = A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  snss  4316  mptelee  25775