Theorem elabgf 1889

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 groups:   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 1459	. . . 4 |- (z e. {x | ph} -> A.x z e. {x | ph})
3	1, 2	hbel 1558	. . 3 |- (A e. {x | ph} -> A.x A e. {x | ph})
4		elabgf.2	. . 3 |- (ps -> A.xps)
5	3, 4	hbbi 1007	. 2 |- ((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps))
6		eleq1 1526	. . 3 |- (x = A -> (x e. {x | ph} <-> A e. {x | ph}))
7		elabgf.3	. . 3 |- (x = A -> (ph <-> ps))
8	6, 7	bibi12d 627	. 2 |- (x = A -> ((x e. {x | ph} <-> ph) <-> (A e. {x | ph} <-> ps)))
9		abid 1458	. 2 |- (x e. {x | ph} <-> ph)
10	1, 5, 8, 9	vtoclgf 1837	1 |- (A e. B -> (A e. {x | ph} <-> ps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 951   = wceq 953   e. wcel 955  {cab 1456

This theorem is referenced by:  elabg 1890  elrabf 1895  cardprc 4833

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803

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