Theorem elabgf 2685

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 2180	. . . 4 |- F/_ x { x | ph }
3	1, 2	nfel 2186	. . 3 |- F/ x A e. { x | ph }
4		elabgf.2	. . 3 |- F/ x ps
5	3, 4	nfbi 1481	. 2 |- F/ x ( A e. { x | ph } <-> ps )
6		eleq1 2100	. . 3 |- ( x = A -> ( x e. { x | ph } <-> A e. { x | ph } ) )
7		elabgf.3	. . 3 |- ( x = A -> ( ph <-> ps ) )
8	6, 7	bibi12d 224	. 2 |- ( x = A -> ( ( x e. { x | ph } <-> ph ) <-> ( A e. { x | ph } <-> ps ) ) )
9		abid 2028	. 2 |- ( x e. { x | ph } <-> ph )
10	1, 5, 8, 9	vtoclgf 2612	1 |- ( A e. B -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   F/wnf 1349   e. wcel 1393   {cab 2026   F/_wnfc 2165

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  elabf  2686  elabg  2688  elab3gf  2692  elrabf  2696  bj-intabssel  9928