Theorem clelab 2587

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 2458	. 2 |- ( A e. { x | ph } <-> E. y ( y = A /\ y e. { x | ph } ) )
2		nfv 1772	. . 3 |- F/ y ( x = A /\ ph )
3		nfv 1772	. . . 4 |- F/ x y = A
4		nfsab1 2452	. . . 4 |- F/ x y e. { x | ph }
5	3, 4	nfan 2022	. . 3 |- F/ x ( y = A /\ y e. { x | ph } )
6		eqeq1 2466	. . . 4 |- ( x = y -> ( x = A <-> y = A ) )
7		sbequ12 2094	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
8		df-clab 2449	. . . . 5 |- ( y e. { x | ph } <-> [ y / x ] ph )
9	7, 8	syl6bbr 271	. . . 4 |- ( x = y -> ( ph <-> y e. { x | ph } ) )
10	6, 9	anbi12d 722	. . 3 |- ( x = y -> ( ( x = A /\ ph ) <-> ( y = A /\ y e. { x | ph } ) ) )
11	2, 5, 10	cbvex 2126	. 2 |- ( E. x ( x = A /\ ph ) <-> E. y ( y = A /\ y e. { x | ph } ) )
12	1, 11	bitr4i 260	1 |- ( A e. { x | ph } <-> E. x ( x = A /\ ph ) )

Syntax hints:   <-> wb 189   /\ wa 375   = wceq 1455   E.wex 1674   [wsb 1808   e. wcel 1898   {cab 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458

This theorem is referenced by:  elrabi  3205  bj-csbsnlem  31550  frege55c  36559