Theorem elabf 1896

Description: Membership in a class abstraction with implicit substitution.

Hypotheses

Ref	Expression
elabf.1	|- (ps -> A.xps)
elabf.2	|- A e. V
elabf.3	|- (x = A -> (ph <-> ps))

Assertion

Ref	Expression
elabf	|- (A e. {x | ph} <-> ps)

Distinct variable group:   x,A

Proof of Theorem elabf

Step	Hyp	Ref	Expression
1		ax-17 971	. . . 4 |- (y e. A -> A.x y e. A)
2		hbab1 1466	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
3	1, 2	hbel 1566	. . 3 |- (A e. {x | ph} -> A.x A e. {x | ph})
4		elabf.1	. . 3 |- (ps -> A.xps)
5	3, 4	hbbi 1010	. 2 |- ((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps))
6		elabf.2	. 2 |- A e. V
7		eleq1 1534	. . . 4 |- (x = A -> (x e. {x | ph} <-> A e. {x | ph}))
8		abid 1465	. . . 4 |- (x e. {x | ph} <-> ph)
9	7, 8	syl5bbr 534	. . 3 |- (x = A -> (ph <-> A e. {x | ph}))
10		elabf.3	. . 3 |- (x = A -> (ph <-> ps))
11	9, 10	bitr3d 530	. 2 |- (x = A -> (A e. {x | ph} <-> ps))
12	5, 6, 11	vtoclef 1857	1 |- (A e. {x | ph} <-> ps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  {cab 1463  Vcvv 1811

This theorem is referenced by:  elab 1897  cbvab 1908  qusp 10555

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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