Theorem elabgft1 13813

Description: One implication of elabgf 2872, 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 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 1448	. . 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 2314	. . . . . 6 |- F/_ x { x | ph }
8	6, 7	nfel 2321	. . . . 5 |- F/ x A e. { x | ph }
9		elabgf1.nf2	. . . . 5 |- F/ x ps
10	8, 9	nfim 1565	. . . 4 |- F/ x ( A e. { x | ph } -> ps )
11		elabgf0 13812	. . . 4 |- ( x = A -> ( A e. { x | ph } <-> ph ) )
12	6, 10, 11	bj-vtoclgft 13810	. . 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 1346   = 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:  elabgf1  13814