Theorem elabgft1 12390

Description: One implication of elabgf 2772, 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		bi1 117	. . . . . 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 1396	. . 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 2237	. . . . . 6 |- F/_ x { x | ph }
8	6, 7	nfel 2244	. . . . 5 |- F/ x A e. { x | ph }
9		elabgf1.nf2	. . . . 5 |- F/ x ps
10	8, 9	nfim 1516	. . . 4 |- F/ x ( A e. { x | ph } -> ps )
11		elabgf0 12389	. . . 4 |- ( x = A -> ( A e. { x | ph } <-> ph ) )
12	6, 10, 11	bj-vtoclgft 12387	. . 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 104   A.wal 1294   = wceq 1296   F/wnf 1401   e. wcel 1445   {cab 2081   F/_wnfc 2222

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635

This theorem is referenced by:  elabgf1  12391