Theorem elabgf 2915

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 2350	. . . 4 |- F/_ x { x | ph }
3	1, 2	nfel 2357	. . 3 |- F/ x A e. { x | ph }
4		elabgf.2	. . 3 |- F/ x ps
5	3, 4	nfbi 1612	. 2 |- F/ x ( A e. { x | ph } <-> ps )
6		eleq1 2268	. . 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 2193	. 2 |- ( x e. { x | ph } <-> ph )
10	1, 5, 8, 9	vtoclgf 2831	1 |- ( A e. B -> ( A e. { x | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   F/wnf 1483   e. wcel 2176   {cab 2191   F/_wnfc 2335

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774

This theorem is referenced by:  elabf  2916  elabg  2919  elab3gf  2923  elrabf  2927  bj-intabssel  15725