Theorem elabgf2 13815

Description: One implication of elabgf 2872. (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 2314	. . . 4 |- F/_ x { x | ph }
4	1, 3	nfel 2321	. . 3 |- F/ x A e. { x | ph }
5	2, 4	nfim 1565	. 2 |- F/ x ( ps -> A e. { x | ph } )
6		elabgf0 13812	. 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 13811	1 |- ( A e. B -> ( ps -> A e. { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   F/wnf 1453   e. wcel 2141   {cab 2156   F/_wnfc 2299

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  elabf2  13817  elabg2  13820  bj-intabssel1  13825