Theorem clelab 1580

Description: Membership of a class variable in a class abstraction.

Assertion

Ref	Expression
clelab	|- (A e. {x | ph} <-> E.x(x = A /\ ph))

Distinct variable group:   x,A

Proof of Theorem clelab

Step	Hyp	Ref	Expression
1		df-clab 1464	. . . 4 |- (y e. {x | ph} <-> [y / x]ph)
2	1	anbi2i 480	. . 3 |- ((y = A /\ y e. {x | ph}) <-> (y = A /\ [y / x]ph))
3	2	exbii 1050	. 2 |- (E.y(y = A /\ y e. {x | ph}) <-> E.y(y = A /\ [y / x]ph))
4		df-clel 1472	. 2 |- (A e. {x | ph} <-> E.y(y = A /\ y e. {x | ph}))
5		ax-17 970	. . 3 |- ((x = A /\ ph) -> A.y(x = A /\ ph))
6		ax-17 970	. . . 4 |- (y = A -> A.x y = A)
7		hbs1 1332	. . . 4 |- ([y / x]ph -> A.x[y / x]ph)
8	6, 7	hban 1008	. . 3 |- ((y = A /\ [y / x]ph) -> A.x(y = A /\ [y / x]ph))
9		eqeq1 1480	. . . 4 |- (x = y -> (x = A <-> y = A))
10		sbequ12 1181	. . . 4 |- (x = y -> (ph <-> [y / x]ph))
11	9, 10	anbi12d 627	. . 3 |- (x = y -> ((x = A /\ ph) <-> (y = A /\ [y / x]ph)))
12	5, 8, 11	cbvex 1165	. 2 |- (E.x(x = A /\ ph) <-> E.y(y = A /\ [y / x]ph))
13	3, 4, 12	3bitr4 183	1 |- (A e. {x | ph} <-> E.x(x = A /\ ph))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  [wsbc 1170  {cab 1463

This theorem is referenced by:  opabid 2807  subtop 7625  bsi 10476

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472

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