Theorem elabgf2 14759

Description: One implication of elabgf 2891. (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 2331	. . . 4 |- F/_ x { x | ph }
4	1, 3	nfel 2338	. . 3 |- F/ x A e. { x | ph }
5	2, 4	nfim 1582	. 2 |- F/ x ( ps -> A e. { x | ph } )
6		elabgf0 14756	. 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 14755	1 |- ( A e. B -> ( ps -> A e. { x | ph } ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1363   F/wnf 1470   e. wcel 2158   {cab 2173   F/_wnfc 2316

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751

This theorem is referenced by:  elabf2  14761  elabg2  14764  bj-intabssel1  14769