Theorem vtoclgft 2598

Description: Closed theorem form of vtoclgf 2606. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)

Assertion

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

Proof of Theorem vtoclgft

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2560	. 2 |- (A e. V -> A e. _V)
2		elisset 2562	. . . . 5 |- (A e. _V -> E.z z = A)
3	2	3ad2ant3 926	. . . 4 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> E.z z = A)
4		nfnfc1 2178	. . . . . . 7 |- F/x F/_xA
5		nfcvd 2176	. . . . . . . 8 |- ( F/_xA -> F/_xz)
6		id 19	. . . . . . . 8 |- ( F/_xA -> F/_xA)
7	5, 6	nfeqd 2189	. . . . . . 7 |- ( F/_xA -> F/x z = A)
8		eqeq1 2043	. . . . . . . 8 |- (z = x -> (z = A <-> x = A))
9	8	a1i 9	. . . . . . 7 |- ( F/_xA -> (z = x -> (z = A <-> x = A)))
10	4, 7, 9	cbvexd 1799	. . . . . 6 |- ( F/_xA -> (E.z z = A <-> E.x x = A))
11	10	ad2antrr 457	. . . . 5 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph)) -> (E.z z = A <-> E.x x = A))
12	11	3adant3 923	. . . 4 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> (E.z z = A <-> E.x x = A))
13	3, 12	mpbid 135	. . 3 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> E.x x = A)
14		bi1 111	. . . . . . . . 9 |- ((ph <-> ps) -> (ph -> ps))
15	14	imim2i 12	. . . . . . . 8 |- ((x = A -> (ph <-> ps)) -> (x = A -> (ph -> ps)))
16	15	com23 72	. . . . . . 7 |- ((x = A -> (ph <-> ps)) -> (ph -> (x = A -> ps)))
17	16	imp 115	. . . . . 6 |- (((x = A -> (ph <-> ps)) /\ ph) -> (x = A -> ps))
18	17	alanimi 1345	. . . . 5 |- ((A.x(x = A -> (ph <-> ps)) /\ A.xph) -> A.x(x = A -> ps))
19	18	3ad2ant2 925	. . . 4 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> A.x(x = A -> ps))
20		simp1r 928	. . . . 5 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> F/xps)
21		19.23t 1564	. . . . 5 |- ( F/xps -> (A.x(x = A -> ps) <-> (E.x x = A -> ps)))
22	20, 21	syl 14	. . . 4 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> (A.x(x = A -> ps) <-> (E.x x = A -> ps)))
23	19, 22	mpbid 135	. . 3 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> (E.x x = A -> ps))
24	13, 23	mpd 13	. 2 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> ps)
25	1, 24	syl3an3 1169	1 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. V) -> ps)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884  A.wal 1240   = wceq 1242   F/wnf 1346  E.wex 1378   e. wcel 1390   F/_wnfc 2162   _Vcvv 2551

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553

This theorem is referenced by:  vtocldf  2599