Theorem clabel 2264

Description: Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)

Assertion

Ref	Expression
clabel	|- ( { x | ph } e. A <-> E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) )

Distinct variable groups:   y, A   ph, y   x, y

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem clabel

Step	Hyp	Ref	Expression
1		df-clel 2133	. 2 |- ( { x | ph } e. A <-> E. y ( y = { x | ph } /\ y e. A ) )
2		abeq2 2246	. . . 4 |- ( y = { x | ph } <-> A. x ( x e. y <-> ph ) )
3	2	anbi2ci 454	. . 3 |- ( ( y = { x | ph } /\ y e. A ) <-> ( y e. A /\ A. x ( x e. y <-> ph ) ) )
4	3	exbii 1584	. 2 |- ( E. y ( y = { x | ph } /\ y e. A ) <-> E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) )
5	1, 4	bitri 183	1 |- ( { x | ph } e. A <-> E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1329   = wceq 1331   E.wex 1468   e. wcel 1480   {cab 2123

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133

This theorem is referenced by:  frecabcl  6289