Theorem elabgf 2984

Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. This version has bound-variable hypotheses in place of distinct variable restrictions. (Contributed by NM, 21-Sep-2003.) (Revised by Mario Carneiro, 12-Oct-2016.)

Hypotheses

Ref	Expression
elabgf.1	|- F/_xA
elabgf.2	|- F/xps
elabgf.3	|- (x = A -> (ph <-> ps))

Assertion

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

Proof of Theorem elabgf

Step	Hyp	Ref	Expression
1		elabgf.1	. 2 |- F/_xA
2		nfab1 2492	. . . 4 |- F/_x{x | ph}
3	1, 2	nfel 2498	. . 3 |- F/x A e. {x | ph}
4		elabgf.2	. . 3 |- F/xps
5	3, 4	nfbi 1834	. 2 |- F/x(A e. {x | ph} <-> ps)
6		eleq1 2413	. . 3 |- (x = A -> (x e. {x | ph} <-> A e. {x | ph}))
7		elabgf.3	. . 3 |- (x = A -> (ph <-> ps))
8	6, 7	bibi12d 312	. 2 |- (x = A -> ((x e. {x | ph} <-> ph) <-> (A e. {x | ph} <-> ps)))
9		abid 2341	. 2 |- (x e. {x | ph} <-> ph)
10	1, 5, 8, 9	vtoclgf 2914	1 |- (A e. B -> (A e. {x | ph} <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544   = wceq 1642   e. wcel 1710  {cab 2339   F/_wnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  elabf  2985  elabg  2987  elab3gf  2991  elrabf  2994