Theorem clabel 2214

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 2085	. 2 |- ( { x | ph } e. A <-> E. y ( y = { x | ph } /\ y e. A ) )
2		abeq2 2197	. . . 4 |- ( y = { x | ph } <-> A. x ( x e. y <-> ph ) )
3	2	anbi2ci 448	. . 3 |- ( ( y = { x | ph } /\ y e. A ) <-> ( y e. A /\ A. x ( x e. y <-> ph ) ) )
4	3	exbii 1542	. 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 1288   = wceq 1290   E.wex 1427   e. wcel 1439   {cab 2075

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-11 1443  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085

This theorem is referenced by:  frecabcl  6180