Theorem elabgf2 14929

Description: One implication of elabgf 2894. (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 2334	. . . 4 |- F/_ x { x | ph }
4	1, 3	nfel 2341	. . 3 |- F/ x A e. { x | ph }
5	2, 4	nfim 1583	. 2 |- F/ x ( ps -> A e. { x | ph } )
6		elabgf0 14926	. 2 |- ( x = A -> ( A e. { x | ph } <-> ph ) )
7		bicom1 131	. . 3 |- ( ( A e. { x | ph } <-> ph ) -> ( ph <-> A e. { x | ph } ) )
8		elabgf2.1	. . . 4 |- ( x = A -> ( ps -> ph ) )
9		biimp 118	. . . 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 14925	1 |- ( A e. B -> ( ps -> A e. { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   F/wnf 1471   e. wcel 2160   {cab 2175   F/_wnfc 2319

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754

This theorem is referenced by:  elabf2  14931  elabg2  14934  bj-intabssel1  14939