Theorem elabgf2 13661

Description: One implication of elabgf 2868. (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 2310	. . . 4 |- F/_ x { x | ph }
4	1, 3	nfel 2317	. . 3 |- F/ x A e. { x | ph }
5	2, 4	nfim 1560	. 2 |- F/ x ( ps -> A e. { x | ph } )
6		elabgf0 13658	. 2 |- ( x = A -> ( A e. { x | ph } <-> ph ) )
7		bicom1 130	. . 3 |- ( ( A e. { x | ph } <-> ph ) -> ( ph <-> A e. { x | ph } ) )
8		elabgf2.1	. . . 4 |- ( x = A -> ( ps -> ph ) )
9		biimp 117	. . . 4 |- ( ( ph <-> A e. { x | ph } ) -> ( ph -> A e. { x | ph } ) )
10	8, 9	syl9 72	. . 3 |- ( x = A -> ( ( ph <-> A e. { x | ph } ) -> ( ps -> A e. { x | ph } ) ) )
11	7, 10	syl5 32	. 2 |- ( x = A -> ( ( A e. { x | ph } <-> ph ) -> ( ps -> A e. { x | ph } ) ) )
12	1, 5, 6, 11	bj-vtoclgf 13657	1 |- ( A e. B -> ( ps -> A e. { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   F/wnf 1448   e. wcel 2136   {cab 2151   F/_wnfc 2295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  elabf2  13663  elabg2  13666  bj-intabssel1  13671