Theorem vtoclgft 2802

Description: Closed theorem form of vtoclgf 2810. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)

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		elex 2763	. 2 |- ( A e. V -> A e. _V )
2		elisset 2766	. . . . 5 |- ( A e. _V -> E. z z = A )
3	2	3ad2ant3 1022	. . . 4 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> E. z z = A )
4		nfnfc1 2335	. . . . . . 7 |- F/ x F/_ x A
5		nfcvd 2333	. . . . . . . 8 |- ( F/_ x A -> F/_ x z )
6		id 19	. . . . . . . 8 |- ( F/_ x A -> F/_ x A )
7	5, 6	nfeqd 2347	. . . . . . 7 |- ( F/_ x A -> F/ x z = A )
8		eqeq1 2196	. . . . . . . 8 |- ( z = x -> ( z = A <-> x = A ) )
9	8	a1i 9	. . . . . . 7 |- ( F/_ x A -> ( z = x -> ( z = A <-> x = A ) ) )
10	4, 7, 9	cbvexd 1939	. . . . . 6 |- ( F/_ x A -> ( E. z z = A <-> E. x x = A ) )
11	10	ad2antrr 488	. . . . 5 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) ) -> ( E. z z = A <-> E. x x = A ) )
12	11	3adant3 1019	. . . 4 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> ( E. z z = A <-> E. x x = A ) )
13	3, 12	mpbid 147	. . 3 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> E. x x = A )
14		biimp 118	. . . . . . . . 9 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
15	14	imim2i 12	. . . . . . . 8 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
16	15	com23 78	. . . . . . 7 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ph -> ( x = A -> ps ) ) )
17	16	imp 124	. . . . . 6 |- ( ( ( x = A -> ( ph <-> ps ) ) /\ ph ) -> ( x = A -> ps ) )
18	17	alanimi 1470	. . . . 5 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) -> A. x ( x = A -> ps ) )
19	18	3ad2ant2 1021	. . . 4 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> A. x ( x = A -> ps ) )
20		simp1r 1024	. . . . 5 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> F/ x ps )
21		19.23t 1688	. . . . 5 |- ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
22	20, 21	syl 14	. . . 4 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
23	19, 22	mpbid 147	. . 3 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> ( E. x x = A -> ps ) )
24	13, 23	mpd 13	. 2 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> ps )
25	1, 24	syl3an3 1284	1 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. V ) -> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   A.wal 1362   = wceq 1364   F/wnf 1471   E.wex 1503   e. wcel 2160   F/_wnfc 2319   _Vcvv 2752

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754

This theorem is referenced by:  vtocldf  2803