Theorem clabel 2333

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 2202	. 2 |- ( { x | ph } e. A <-> E. y ( y = { x | ph } /\ y e. A ) )
2		abeq2 2315	. . . 4 |- ( y = { x | ph } <-> A. x ( x e. y <-> ph ) )
3	2	anbi2ci 459	. . 3 |- ( ( y = { x | ph } /\ y e. A ) <-> ( y e. A /\ A. x ( x e. y <-> ph ) ) )
4	3	exbii 1629	. 2 |- ( E. y ( y = { x | ph } /\ y e. A ) <-> E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) )
5	1, 4	bitri 184	1 |- ( { x | ph } e. A <-> E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2177   {cab 2192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202

This theorem is referenced by:  frecabcl  6503