Theorem clelab 2296

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 2157	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
2	1	anbi2i 454	. . 3 |- ( ( y = A /\ y e. { x | ph } ) <-> ( y = A /\ [ y / x ] ph ) )
3	2	exbii 1598	. 2 |- ( E. y ( y = A /\ y e. { x | ph } ) <-> E. y ( y = A /\ [ y / x ] ph ) )
4		df-clel 2166	. 2 |- ( A e. { x | ph } <-> E. y ( y = A /\ y e. { x | ph } ) )
5		nfv 1521	. . 3 |- F/ y ( x = A /\ ph )
6		nfv 1521	. . . 4 |- F/ x y = A
7		nfs1v 1932	. . . 4 |- F/ x [ y / x ] ph
8	6, 7	nfan 1558	. . 3 |- F/ x ( y = A /\ [ y / x ] ph )
9		eqeq1 2177	. . . 4 |- ( x = y -> ( x = A <-> y = A ) )
10		sbequ12 1764	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
11	9, 10	anbi12d 470	. . 3 |- ( x = y -> ( ( x = A /\ ph ) <-> ( y = A /\ [ y / x ] ph ) ) )
12	5, 8, 11	cbvex 1749	. 2 |- ( E. x ( x = A /\ ph ) <-> E. y ( y = A /\ [ y / x ] ph ) )
13	3, 4, 12	3bitr4i 211	1 |- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1348   E.wex 1485   [wsb 1755   e. wcel 2141   {cab 2156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166

This theorem is referenced by:  elrabi  2883