Theorem clelab 2291

Description: Membership of a class variable in a class abstraction.

Assertion

Ref	Expression
clelab	|- (A e. {x | ph} <-> E.x(x = A /\ ph))

Distinct variable group:   x,A

Proof of Theorem clelab

Step	Hyp	Ref	Expression
1		df-clab 2158	. . . 4 |- (y e. {x | ph} <-> [y / x]ph)
2	1	anbi2i 804	. . 3 |- ((y = A /\ y e. {x | ph}) <-> (y = A /\ [y / x]ph))
3	2	exbii 1716	. 2 |- (E.y(y = A /\ y e. {x | ph}) <-> E.y(y = A /\ [y / x]ph))
4		df-clel 2166	. 2 |- (A e. {x | ph} <-> E.y(y = A /\ y e. {x | ph}))
5		ax-17 1634	. . 3 |- ((x = A /\ ph) -> A.y(x = A /\ ph))
6		ax-17 1634	. . . 4 |- (y = A -> A.x y = A)
7		hbs1 2014	. . . 4 |- ([y / x]ph -> A.x[y / x]ph)
8	6, 7	hban 1674	. . 3 |- ((y = A /\ [y / x]ph) -> A.x(y = A /\ [y / x]ph))
9		eqeq1 2176	. . . 4 |- (x = y -> (x = A <-> y = A))
10		sbequ12 1854	. . . 4 |- (x = y -> (ph <-> [y / x]ph))
11	9, 10	anbi12d 821	. . 3 |- (x = y -> ((x = A /\ ph) <-> (y = A /\ [y / x]ph)))
12	5, 8, 11	cbvex 1838	. 2 |- (E.x(x = A /\ ph) <-> E.y(y = A /\ [y / x]ph))
13	3, 4, 12	3bitr4i 340	1 |- (A e. {x | ph} <-> E.x(x = A /\ ph))

Syntax hints:   <-> wb 231   /\ wa 433   = wceq 1615   e. wcel 1617  E.wex 1644  [wsbc 1843  {cab 2157

This theorem is referenced by:  subtop 9917  bsi 15859  intopcoaconb 15923

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-10 1625  ax-12 1627  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893  ax-ext 2152

This theorem depends on definitions:  df-bi 232  df-an 435  df-ex 1645  df-sb 1845  df-clab 2158  df-cleq 2163  df-clel 2166

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