Theorem clabel 2297

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 2166	. 2 |- ( { x | ph } e. A <-> E. y ( y = { x | ph } /\ y e. A ) )
2		abeq2 2279	. . . 4 |- ( y = { x | ph } <-> A. x ( x e. y <-> ph ) )
3	2	anbi2ci 456	. . 3 |- ( ( y = { x | ph } /\ y e. A ) <-> ( y e. A /\ A. x ( x e. y <-> ph ) ) )
4	3	exbii 1598	. 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 1346   = wceq 1348   E.wex 1485   e. wcel 2141   {cab 2156

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166

This theorem is referenced by:  frecabcl  6378