Theorem elabgf 1901

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 1469	. . . 4 |- (z e. {x | ph} -> A.x z e. {x | ph})
3	1, 2	hbel 1569	. . 3 |- (A e. {x | ph} -> A.x A e. {x | ph})
4		elabgf.2	. . 3 |- (ps -> A.xps)
5	3, 4	hbbi 1012	. 2 |- ((A e. {x | ph} <-> ps) -> A.x(A e. {x | ph} <-> ps))
6		eleq1 1537	. . 3 |- (x = A -> (x e. {x | ph} <-> A e. {x | ph}))
7		elabgf.3	. . 3 |- (x = A -> (ph <-> ps))
8	6, 7	bibi12d 631	. 2 |- (x = A -> ((x e. {x | ph} <-> ph) <-> (A e. {x | ph} <-> ps)))
9		abid 1468	. 2 |- (x e. {x | ph} <-> ph)
10	1, 5, 8, 9	vtoclgf 1849	1 |- (A e. B -> (A e. {x | ph} <-> ps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 956   = wceq 958   e. wcel 960  {cab 1466

This theorem is referenced by:  elabg 1902  elrabf 1907  cardprc 4872

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815

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