Theorem vtoclgft 3254

Description: Closed theorem form of vtoclgf 3264. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by JJ, 11-Aug-2021.)

Assertion

Ref	Expression
vtoclgft	|- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )

Proof of Theorem vtoclgft

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elisset 3215	. . . . 5 |- ( A e. V -> E. z z = A )
2		nfnfc1 2767	. . . . . 6 |- F/ x F/_ x A
3		nfcvd 2765	. . . . . . 7 |- ( F/_ x A -> F/_ x z )
4		id 22	. . . . . . 7 |- ( F/_ x A -> F/_ x A )
5	3, 4	nfeqd 2772	. . . . . 6 |- ( F/_ x A -> F/ x z = A )
6		eqeq1 2626	. . . . . . 7 |- ( z = x -> ( z = A <-> x = A ) )
7	6	a1i 11	. . . . . 6 |- ( F/_ x A -> ( z = x -> ( z = A <-> x = A ) ) )
8	2, 5, 7	cbvexd 2278	. . . . 5 |- ( F/_ x A -> ( E. z z = A <-> E. x x = A ) )
9	1, 8	syl5ib 234	. . . 4 |- ( F/_ x A -> ( A e. V -> E. x x = A ) )
10	9	ad2antrr 762	. . 3 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) ) -> ( A e. V -> E. x x = A ) )
11	10	3impia 1261	. 2 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> E. x x = A )
12		biimp 205	. . . . . . . . 9 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
13	12	imim2i 16	. . . . . . . 8 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
14	13	com23 86	. . . . . . 7 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ph -> ( x = A -> ps ) ) )
15	14	imp 445	. . . . . 6 |- ( ( ( x = A -> ( ph <-> ps ) ) /\ ph ) -> ( x = A -> ps ) )
16	15	alanimi 1744	. . . . 5 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) -> A. x ( x = A -> ps ) )
17		19.23t 2079	. . . . . 6 |- ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
18	17	adantl 482	. . . . 5 |- ( ( F/_ x A /\ F/ x ps ) -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
19	16, 18	syl5ib 234	. . . 4 |- ( ( F/_ x A /\ F/ x ps ) -> ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) -> ( E. x x = A -> ps ) ) )
20	19	imp 445	. . 3 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) ) -> ( E. x x = A -> ps ) )
21	20	3adant3 1081	. 2 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ( E. x x = A -> ps ) )
22	11, 21	mpd 15	1 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   F/wnf 1708   e. wcel 1990   F/_wnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  vtocldf  3256  bj-vtoclgfALT  33021