Theorem elabgft1 16100

Description: One implication of elabgf 2945, in closed form. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabgf1.nf1	|- F/_ x A
elabgf1.nf2	|- F/ x ps

Assertion

Ref	Expression
elabgft1	|- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )

Proof of Theorem elabgft1

Step	Hyp	Ref	Expression
1		biimp 118	. . . . . 6 |- ( ( A e. { x | ph } <-> ph ) -> ( A e. { x | ph } -> ph ) )
2		imim2 55	. . . . . 6 |- ( ( ph -> ps ) -> ( ( A e. { x | ph } -> ph ) -> ( A e. { x | ph } -> ps ) ) )
3	1, 2	syl5 32	. . . . 5 |- ( ( ph -> ps ) -> ( ( A e. { x | ph } <-> ph ) -> ( A e. { x | ph } -> ps ) ) )
4	3	imim2i 12	. . . 4 |- ( ( x = A -> ( ph -> ps ) ) -> ( x = A -> ( ( A e. { x | ph } <-> ph ) -> ( A e. { x | ph } -> ps ) ) ) )
5	4	alimi 1501	. . 3 |- ( A. x ( x = A -> ( ph -> ps ) ) -> A. x ( x = A -> ( ( A e. { x | ph } <-> ph ) -> ( A e. { x | ph } -> ps ) ) ) )
6		elabgf1.nf1	. . . 4 |- F/_ x A
7		nfab1 2374	. . . . . 6 |- F/_ x { x | ph }
8	6, 7	nfel 2381	. . . . 5 |- F/ x A e. { x | ph }
9		elabgf1.nf2	. . . . 5 |- F/ x ps
10	8, 9	nfim 1618	. . . 4 |- F/ x ( A e. { x | ph } -> ps )
11		elabgf0 16099	. . . 4 |- ( x = A -> ( A e. { x | ph } <-> ph ) )
12	6, 10, 11	bj-vtoclgft 16097	. . 3 |- ( A. x ( x = A -> ( ( A e. { x | ph } <-> ph ) -> ( A e. { x | ph } -> ps ) ) ) -> ( A e. { x | ph } -> ( A e. { x | ph } -> ps ) ) )
13	5, 12	syl 14	. 2 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ( A e. { x | ph } -> ps ) ) )
14	13	pm2.43d 50	1 |- ( A. x ( x = A -> ( ph -> ps ) ) -> ( A e. { x | ph } -> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1393   = wceq 1395   F/wnf 1506   e. wcel 2200   {cab 2215   F/_wnfc 2359

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  elabgf1  16101