Theorem clel2 2859

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 2229	. . 3 |- ( x = A -> ( x e. B <-> A e. B ) )
3	1, 2	ceqsalv 2756	. 2 |- ( A. x ( x = A -> x e. B ) <-> A e. B )
4	3	bicomi 131	1 |- ( A e. B <-> A. x ( x = A -> x e. B ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   e. wcel 2136   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2728

This theorem is referenced by:  snss  3702