Theorem elabf 1892

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-17 969	. . . 4 |- (y e. A -> A.x y e. A)
2		hbab1 1464	. . . 4 |- (y e. {x | ph} -> A.x y e. {x | ph})
3	1, 2	hbel 1563	. . 3 |- (A e. {x | ph} -> A.x A e. {x | ph})
4		elabf.1	. . 3 |- (ps -> A.xps)
5	3, 4	hbbi 1008	. 2 |- ((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps))
6		elabf.2	. 2 |- A e. V
7		eleq1 1531	. . . 4 |- (x = A -> (x e. {x | ph} <-> A e. {x | ph}))
8		abid 1463	. . . 4 |- (x e. {x | ph} <-> ph)
9	7, 8	syl5bbr 533	. . 3 |- (x = A -> (ph <-> A e. {x | ph}))
10		elabf.3	. . 3 |- (x = A -> (ph <-> ps))
11	9, 10	bitr3d 529	. 2 |- (x = A -> (A e. {x | ph} <-> ps))
12	5, 6, 11	vtoclef 1853	1 |- (A e. {x | ph} <-> ps)

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952   = wceq 954   e. wcel 956  {cab 1461  Vcvv 1807

This theorem is referenced by:  elab 1893  cbvab 1904  qusp 10466

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808

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