Theorem clel2 2976

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 2413	. . 3 |- (x = A -> (x e. B <-> A e. B))
3	1, 2	ceqsalv 2886	. 2 |- (A.x(x = A -> x e. B) <-> A e. B)
4	3	bicomi 193	1 |- (A e. B <-> A.x(x = A -> x e. B))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  snss  3839