Theorem clelab 2357

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 2218	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
2	1	anbi2i 457	. . 3 |- ( ( y = A /\ y e. { x | ph } ) <-> ( y = A /\ [ y / x ] ph ) )
3	2	exbii 1653	. 2 |- ( E. y ( y = A /\ y e. { x | ph } ) <-> E. y ( y = A /\ [ y / x ] ph ) )
4		df-clel 2227	. 2 |- ( A e. { x | ph } <-> E. y ( y = A /\ y e. { x | ph } ) )
5		nfv 1576	. . 3 |- F/ y ( x = A /\ ph )
6		nfv 1576	. . . 4 |- F/ x y = A
7		nfs1v 1992	. . . 4 |- F/ x [ y / x ] ph
8	6, 7	nfan 1613	. . 3 |- F/ x ( y = A /\ [ y / x ] ph )
9		eqeq1 2238	. . . 4 |- ( x = y -> ( x = A <-> y = A ) )
10		sbequ12 1819	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
11	9, 10	anbi12d 473	. . 3 |- ( x = y -> ( ( x = A /\ ph ) <-> ( y = A /\ [ y / x ] ph ) ) )
12	5, 8, 11	cbvex 1804	. 2 |- ( E. x ( x = A /\ ph ) <-> E. y ( y = A /\ [ y / x ] ph ) )
13	3, 4, 12	3bitr4i 212	1 |- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1397   E.wex 1540   [wsb 1810   e. wcel 2202   {cab 2217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227

This theorem is referenced by:  elrabi  2959