Theorem elabgf 1416

Description: Membership in a class abstraction with implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound variable hypotheses in place of distinct variable restrictions.

Hypotheses

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

Assertion

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

Distinct variable group(s):   y,A   x,y

Proof of Theorem elabgf

Step	Hyp	Ref	Expression
1		elabgf.1	. 2 |- (y e. A -> A.x y e. A)
2		hbab1 1095	. . . 4 |- (z e. {x | ph} -> A.x z e. {x | ph})
3	1, 2	hbel 1172	. . 3 |- (A e. {x | ph} -> A.x A e. {x | ph})
4		elabgf.2	. . 3 |- (ps -> A.xps)
5	3, 4	hbbi 705	. 2 |- ((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps))
6		eleq1 1149	. . 3 |- (x = A -> (x e. {x | ph} <-> A e. {x | ph}))
7		elabgf.3	. . 3 |- (x = A -> (ph <-> ps))
8	6, 7	bibi12d 477	. 2 |- (x = A -> ((x e. {x | ph} <-> ph) <-> (A e. {x | ph} <-> ps)))
9		abid 1094	. 2 |- (x e. {x | ph} <-> ph)
10	1, 5, 8, 9	vtoclgf 1382	1 |- (A e. B -> (A e. {x | ph} <-> ps))

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

This theorem is referenced by:  elabg 1417  elrabf 1421  cardprc 3667

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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