Theorem clelab 1187

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(s):   x,A

Proof of Theorem clelab

Step	Hyp	Ref	Expression
1		df-clab 1093	. . . 4 |- (y e. {x | ph} <-> [y / x]ph)
2	1	anbi2i 367	. . 3 |- ((y = A /\ y e. {x | ph}) <-> (y = A /\ [y / x]ph))
3	2	biex 733	. 2 |- (E.y(y = A /\ y e. {x | ph}) <-> E.y(y = A /\ [y / x]ph))
4		df-clel 1099	. 2 |- (A e. {x | ph} <-> E.y(y = A /\ y e. {x | ph}))
5		ax-17 925	. . 3 |- ((x = A /\ ph) -> A.y(x = A /\ ph))
6		ax-17 925	. . . 4 |- (y = A -> A.x y = A)
7		hbs1 986	. . . 4 |- ([y / x]ph -> A.x[y / x]ph)
8	6, 7	hban 704	. . 3 |- ((y = A /\ [y / x]ph) -> A.x(y = A /\ [y / x]ph))
9		cleq1 1107	. . . 4 |- (x = y -> (x = A <-> y = A))
10		sbequ12 865	. . . 4 |- (x = y -> (ph <-> [y / x]ph))
11	9, 10	anbi12d 476	. . 3 |- (x = y -> ((x = A /\ ph) <-> (y = A /\ [y / x]ph)))
12	5, 8, 11	cbvex 849	. 2 |- (E.x(x = A /\ ph) <-> E.y(y = A /\ [y / x]ph))
13	3, 4, 12	3bitr4 158	1 |- (A e. {x | ph} <-> E.x(x = A /\ ph))

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678   = weq 797  [wsb 852  {cab 1090   = wceq 1091   e. wcel 1092

This theorem is referenced by:  opabid 2099

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099

metamath.org