Theorem clabel 2292

Description: Membership of a class abstraction in another class.

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

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 2277	. . . . 5 |- (y = {x | ph} <-> A.x(x e. y <-> ph))
3	2	anbi1i 805	. . . 4 |- ((y = {x | ph} /\ y e. A) <-> (A.x(x e. y <-> ph) /\ y e. A))
4		ancom 510	. . . 4 |- ((A.x(x e. y <-> ph) /\ y e. A) <-> (y e. A /\ A.x(x e. y <-> ph)))
5	3, 4	bitri 306	. . 3 |- ((y = {x | ph} /\ y e. A) <-> (y e. A /\ A.x(x e. y <-> ph)))
6	5	exbii 1716	. 2 |- (E.y(y = {x | ph} /\ y e. A) <-> E.y(y e. A /\ A.x(x e. y <-> ph)))
7	1, 6	bitri 306	1 |- ({x | ph} e. A <-> E.y(y e. A /\ A.x(x e. y <-> ph)))

Syntax hints:   <-> wb 231   /\ wa 433  A.wal 1613   = wceq 1615   e. wcel 1617  E.wex 1644  {cab 2157

This theorem is referenced by:  grothprimlem 11169

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-10 1625  ax-12 1627  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893  ax-ext 2152

This theorem depends on definitions:  df-bi 232  df-an 435  df-ex 1645  df-sb 1845  df-clab 2158  df-cleq 2163  df-clel 2166

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