Theorem clabel 1558

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 1449	. 2 |- ({x | ph} e. A <-> E.y(y = {x | ph} /\ y e. A))
2		abeq2 1544	. . . . 5 |- (y = {x | ph} <-> A.x(x e. y <-> ph))
3	2	anbi1i 480	. . . 4 |- ((y = {x | ph} /\ y e. A) <-> (A.x(x e. y <-> ph) /\ y e. A))
4		ancom 435	. . . 4 |- ((A.x(x e. y <-> ph) /\ y e. A) <-> (y e. A /\ A.x(x e. y <-> ph)))
5	3, 4	bitr 173	. . 3 |- ((y = {x | ph} /\ y e. A) <-> (y e. A /\ A.x(x e. y <-> ph)))
6	5	exbii 1027	. 2 |- (E.y(y = {x | ph} /\ y e. A) <-> E.y(y e. A /\ A.x(x e. y <-> ph)))
7	1, 6	bitr 173	1 |- ({x | ph} e. A <-> E.y(y e. A /\ A.x(x e. y <-> ph)))

Syntax hints:   <-> wb 146   /\ wa 223  A.wal 950  E.wex 956   = wceq 1099   e. wcel 1105  {cab 1440

This theorem is referenced by:  grothprimlem 8634

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449

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