Theorem elabgf 2906

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/_ x A
elabgf.2	|- F/ x ps
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/_ x A
2		nfab1 2341	. . . 4 |- F/_ x { x | ph }
3	1, 2	nfel 2348	. . 3 |- F/ x A e. { x | ph }
4		elabgf.2	. . 3 |- F/ x ps
5	3, 4	nfbi 1603	. 2 |- F/ x ( A e. { x | ph } <-> ps )
6		eleq1 2259	. . 3 |- ( x = A -> ( x e. { x | ph } <-> A e. { x | ph } ) )
7		elabgf.3	. . 3 |- ( x = A -> ( ph <-> ps ) )
8	6, 7	bibi12d 235	. 2 |- ( x = A -> ( ( x e. { x | ph } <-> ph ) <-> ( A e. { x | ph } <-> ps ) ) )
9		abid 2184	. 2 |- ( x e. { x | ph } <-> ph )
10	1, 5, 8, 9	vtoclgf 2822	1 |- ( A e. B -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   F/wnf 1474   e. wcel 2167   {cab 2182   F/_wnfc 2326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  elabf  2907  elabg  2910  elab3gf  2914  elrabf  2918  bj-intabssel  15435