Theorem clelab 2162

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 2027	. . . 4 |- ( y e. { x | ph } <-> [ y / x ] ph )
2	1	anbi2i 430	. . 3 |- ( ( y = A /\ y e. { x | ph } ) <-> ( y = A /\ [ y / x ] ph ) )
3	2	exbii 1496	. 2 |- ( E. y ( y = A /\ y e. { x | ph } ) <-> E. y ( y = A /\ [ y / x ] ph ) )
4		df-clel 2036	. 2 |- ( A e. { x | ph } <-> E. y ( y = A /\ y e. { x | ph } ) )
5		nfv 1421	. . 3 |- F/ y ( x = A /\ ph )
6		nfv 1421	. . . 4 |- F/ x y = A
7		nfs1v 1815	. . . 4 |- F/ x [ y / x ] ph
8	6, 7	nfan 1457	. . 3 |- F/ x ( y = A /\ [ y / x ] ph )
9		eqeq1 2046	. . . 4 |- ( x = y -> ( x = A <-> y = A ) )
10		sbequ12 1654	. . . 4 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
11	9, 10	anbi12d 442	. . 3 |- ( x = y -> ( ( x = A /\ ph ) <-> ( y = A /\ [ y / x ] ph ) ) )
12	5, 8, 11	cbvex 1639	. 2 |- ( E. x ( x = A /\ ph ) <-> E. y ( y = A /\ [ y / x ] ph ) )
13	3, 4, 12	3bitr4i 201	1 |- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   [wsb 1645   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036

This theorem is referenced by:  elrabi  2695