Theorem elabgf2 9919

Description: One implication of elabgf 2685. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabgf2.nf1	|- F/_ x A
elabgf2.nf2	|- F/ x ps
elabgf2.1	|- ( x = A -> ( ps -> ph ) )

Assertion

Ref	Expression
elabgf2	|- ( A e. B -> ( ps -> A e. { x | ph } ) )

Proof of Theorem elabgf2

Step	Hyp	Ref	Expression
1		elabgf2.nf1	. 2 |- F/_ x A
2		elabgf2.nf2	. . 3 |- F/ x ps
3		nfab1 2180	. . . 4 |- F/_ x { x | ph }
4	1, 3	nfel 2186	. . 3 |- F/ x A e. { x | ph }
5	2, 4	nfim 1464	. 2 |- F/ x ( ps -> A e. { x | ph } )
6		elabgf0 9916	. 2 |- ( x = A -> ( A e. { x | ph } <-> ph ) )
7		bicom1 122	. . 3 |- ( ( A e. { x | ph } <-> ph ) -> ( ph <-> A e. { x | ph } ) )
8		elabgf2.1	. . . 4 |- ( x = A -> ( ps -> ph ) )
9		bi1 111	. . . 4 |- ( ( ph <-> A e. { x | ph } ) -> ( ph -> A e. { x | ph } ) )
10	8, 9	syl9 66	. . 3 |- ( x = A -> ( ( ph <-> A e. { x | ph } ) -> ( ps -> A e. { x | ph } ) ) )
11	7, 10	syl5 28	. 2 |- ( x = A -> ( ( A e. { x | ph } <-> ph ) -> ( ps -> A e. { x | ph } ) ) )
12	1, 5, 6, 11	bj-vtoclgf 9915	1 |- ( A e. B -> ( ps -> A e. { x | ph } ) )

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:  elabf2  9921  elabg2  9924  bj-intabssel1  9929