Theorem elabf 1414

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(s):   x,A

Proof of Theorem elabf

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 |- (y e. A -> A.x y e. A)
2		hbab1 1095	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
3	1, 2	hbel 1172	. . 3 |- (A e. {x | ph} -> A.x A e. {x | ph})
4		elabf.1	. . 3 |- (ps -> A.xps)
5	3, 4	hbbi 705	. 2 |- ((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps))
6		elabf.2	. 2 |- A e. V
7		eleq1 1149	. . . 4 |- (x = A -> (x e. {x | ph} <-> A e. {x | ph}))
8		abid 1094	. . . 4 |- (x e. {x | ph} <-> ph)
9	7, 8	syl5bbr 412	. . 3 |- (x = A -> (ph <-> A e. {x | ph}))
10		elabf.3	. . 3 |- (x = A -> (ph <-> ps))
11	9, 10	bitr3d 408	. 2 |- (x = A -> (A e. {x | ph} <-> ps))
12	5, 6, 11	vtoclef 1392	1 |- (A e. {x | ph} <-> ps)

Syntax hints:   -> wi 2   <-> wb 127  A.wal 672  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348

This theorem is referenced by:  elab 1415  cbvab 1423

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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