Theorem clelab 2748

Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)

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-clel 2618	. 2 |- ( A e. { x | ph } <-> E. y ( y = A /\ y e. { x | ph } ) )
2		nfv 1843	. . 3 |- F/ y ( x = A /\ ph )
3		nfv 1843	. . . 4 |- F/ x y = A
4		nfsab1 2612	. . . 4 |- F/ x y e. { x | ph }
5	3, 4	nfan 1828	. . 3 |- F/ x ( y = A /\ y e. { x | ph } )
6		eqeq1 2626	. . . 4 |- ( x = y -> ( x = A <-> y = A ) )
7		sbequ12 2111	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		df-clab 2609	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
9	7, 8	syl6bbr 278	. . . 4 |- ( x = y -> ( ph <-> y e. { x | ph } ) )
10	6, 9	anbi12d 747	. . 3 |- ( x = y -> ( ( x = A /\ ph ) <-> ( y = A /\ y e. { x | ph } ) ) )
11	2, 5, 10	cbvex 2272	. 2 |- ( E. x ( x = A /\ ph ) <-> E. y ( y = A /\ y e. { x | ph } ) )
12	1, 11	bitr4i 267	1 |- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   {cab 2608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618

This theorem is referenced by:  elrabi  3359  bj-csbsnlem  32898  frege55c  38212  spr0nelg  41726