Theorem clabel 2014

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 1880	. 2 |- ({x | ph} e. A <-> E.y(y = {x | ph} /\ y e. A))
2		abeq2 1999	. . . . 5 |- (y = {x | ph} <-> A.x(x e. y <-> ph))
3	2	anbi1i 539	. . . 4 |- ((y = {x | ph} /\ y e. A) <-> (A.x(x e. y <-> ph) /\ y e. A))
4		ancom 482	. . . 4 |- ((A.x(x e. y <-> ph) /\ y e. A) <-> (y e. A /\ A.x(x e. y <-> ph)))
5	3, 4	bitri 190	. . 3 |- ((y = {x | ph} /\ y e. A) <-> (y e. A /\ A.x(x e. y <-> ph)))
6	5	exbii 1398	. 2 |- (E.y(y = {x | ph} /\ y e. A) <-> E.y(y e. A /\ A.x(x e. y <-> ph)))
7	1, 6	bitri 190	1 |- ({x | ph} e. A <-> E.y(y e. A /\ A.x(x e. y <-> ph)))

Syntax hints:   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871

This theorem is referenced by:  grothprimlem 10140

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880

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