Theorem vtoclgft 3008

Description: Closed theorem form of vtoclgf 3016. (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 2970	. 2 |- ( A e. V -> A e. _V )
2		elisset 2972	. . . . 5 |- ( A e. _V -> E. z z = A )
3	2	3ad2ant3 981	. . . 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 2581	. . . . . . 7 |- F/ x F/_ x A
5		nfcvd 2579	. . . . . . . 8 |- ( F/_ x A -> F/_ x z )
6		id 21	. . . . . . . 8 |- ( F/_ x A -> F/_ x A )
7	5, 6	nfeqd 2592	. . . . . . 7 |- ( F/_ x A -> F/ x z = A )
8		eqeq1 2448	. . . . . . . 8 |- ( z = x -> ( z = A <-> x = A ) )
9	8	a1i 11	. . . . . . 7 |- ( F/_ x A -> ( z = x -> ( z = A <-> x = A ) ) )
10	4, 7, 9	cbvexd 1991	. . . . . 6 |- ( F/_ x A -> ( E. z z = A <-> E. x x = A ) )
11	10	ad2antrr 708	. . . . 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 978	. . . 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 203	. . 3 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> E. x x = A )
14		bi1 180	. . . . . . . . 9 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
15	14	imim2i 14	. . . . . . . 8 |- ( ( x = A -> ( ph <-> ps ) ) -> ( x = A -> ( ph -> ps ) ) )
16	15	com23 75	. . . . . . 7 |- ( ( x = A -> ( ph <-> ps ) ) -> ( ph -> ( x = A -> ps ) ) )
17	16	imp 420	. . . . . 6 |- ( ( ( x = A -> ( ph <-> ps ) ) /\ ph ) -> ( x = A -> ps ) )
18	17	alanimi 1572	. . . . 5 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) -> A. x ( x = A -> ps ) )
19	18	3ad2ant2 980	. . . 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 983	. . . . 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 1820	. . . . 5 |- ( F/ x ps -> ( A. x ( x = A -> ps ) <-> ( E. x x = A -> ps ) ) )
22	20, 21	syl 16	. . . 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 203	. . 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 15	. 2 |- ( ( ( F/_ x A /\ F/ x ps ) /\ ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) /\ A e. _V ) -> ps )
25	1, 24	syl3an3 1220	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 178   /\ wa 360   /\ w3a 937   A.wal 1550   E.wex 1551   F/wnf 1554   = wceq 1653   e. wcel 1727   F/_wnfc 2565   _Vcvv 2962

This theorem is referenced by:  vtocldf  3009  riotasv2dOLD  6624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423

This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-v 2964