Theorem clelab 2203

Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem clelab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clab 2068	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
2	1	anbi2i 444	. . 3 |- ( ( y = A /\ y e. { x | ph } ) <-> ( y = A /\ [ y / x ] ph ) )
3	2	exbii 1536	. 2 |- ( E. y ( y = A /\ y e. { x | ph } ) <-> E. y ( y = A /\ [ y / x ] ph ) )
4		df-clel 2077	. 2 |- ( A e. { x | ph } <-> E. y ( y = A /\ y e. { x | ph } ) )
5		nfv 1461	. . 3 |- F/ y ( x = A /\ ph )
6		nfv 1461	. . . 4 |- F/ x y = A
7		nfs1v 1856	. . . 4 |- F/ x [ y / x ] ph
8	6, 7	nfan 1497	. . 3 |- F/ x ( y = A /\ [ y / x ] ph )
9		eqeq1 2087	. . . 4 |- ( x = y -> ( x = A <-> y = A ) )
10		sbequ12 1694	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
11	9, 10	anbi12d 456	. . 3 |- ( x = y -> ( ( x = A /\ ph ) <-> ( y = A /\ [ y / x ] ph ) ) )
12	5, 8, 11	cbvex 1679	. 2 |- ( E. x ( x = A /\ ph ) <-> E. y ( y = A /\ [ y / x ] ph ) )
13	3, 4, 12	3bitr4i 210	1 |- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   [wsb 1685   {cab 2067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077

This theorem is referenced by:  elrabi  2746