Theorem clelab 2474

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 2340	. . . 4 |- (y e. {x | ph} <-> [y / x]ph)
2	1	anbi2i 675	. . 3 |- ((y = A /\ y e. {x | ph}) <-> (y = A /\ [y / x]ph))
3	2	exbii 1582	. 2 |- (E.y(y = A /\ y e. {x | ph}) <-> E.y(y = A /\ [y / x]ph))
4		df-clel 2349	. 2 |- (A e. {x | ph} <-> E.y(y = A /\ y e. {x | ph}))
5		nfv 1619	. . 3 |- F/y(x = A /\ ph)
6		nfv 1619	. . . 4 |- F/x y = A
7		nfs1v 2106	. . . 4 |- F/x[y / x]ph
8	6, 7	nfan 1824	. . 3 |- F/x(y = A /\ [y / x]ph)
9		eqeq1 2359	. . . 4 |- (x = y -> (x = A <-> y = A))
10		sbequ12 1919	. . . 4 |- (x = y -> (ph <-> [y / x]ph))
11	9, 10	anbi12d 691	. . 3 |- (x = y -> ((x = A /\ ph) <-> (y = A /\ [y / x]ph)))
12	5, 8, 11	cbvex 1985	. 2 |- (E.x(x = A /\ ph) <-> E.y(y = A /\ [y / x]ph))
13	3, 4, 12	3bitr4i 268	1 |- (A e. {x | ph} <-> E.x(x = A /\ ph))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642  [wsb 1648   e. wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349

This theorem is referenced by: (None)