Theorem clabel 2163

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 2036	. 2 |- ( { x | ph } e. A <-> E. y ( y = { x | ph } /\ y e. A ) )
2		abeq2 2146	. . . 4 |- ( y = { x | ph } <-> A. x ( x e. y <-> ph ) )
3	2	anbi2ci 432	. . 3 |- ( ( y = { x | ph } /\ y e. A ) <-> ( y e. A /\ A. x ( x e. y <-> ph ) ) )
4	3	exbii 1496	. 2 |- ( E. y ( y = { x | ph } /\ y e. A ) <-> E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) )
5	1, 4	bitri 173	1 |- ( { x | ph } e. A <-> E. y ( y e. A /\ A. x ( x e. y <-> ph ) ) )

Syntax hints:   /\ wa 97   <-> wb 98   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036

This theorem is referenced by: (None)