Theorem clabel 1582

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 1472	. 2 |- ({x | ph} e. A <-> E.y(y = {x | ph} /\ y e. A))
2		abeq2 1568	. . . . 5 |- (y = {x | ph} <-> A.x(x e. y <-> ph))
3	2	anbi1i 481	. . . 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 1051	. 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 954   = wceq 956   e. wcel 958  E.wex 980  {cab 1463

This theorem is referenced by:  grothprimlem 8782

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472

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