Theorem vtoclgft 2906

Description: Closed theorem form of vtoclgf 2914. (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 2868	. 2 |- (A e. V -> A e. _V)
2		elisset 2870	. . . . 5 |- (A e. _V -> E.z z = A)
3	2	3ad2ant3 978	. . . 4 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> E.z z = A)
4		nfnfc1 2493	. . . . . . 7 |- F/x F/_xA
5		nfcvd 2491	. . . . . . . 8 |- ( F/_xA -> F/_xz)
6		id 19	. . . . . . . 8 |- ( F/_xA -> F/_xA)
7	5, 6	nfeqd 2504	. . . . . . 7 |- ( F/_xA -> F/x z = A)
8		eqeq1 2359	. . . . . . . 8 |- (z = x -> (z = A <-> x = A))
9	8	a1i 10	. . . . . . 7 |- ( F/_xA -> (z = x -> (z = A <-> x = A)))
10	4, 7, 9	cbvexd 2009	. . . . . 6 |- ( F/_xA -> (E.z z = A <-> E.x x = A))
11	10	ad2antrr 706	. . . . 5 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph)) -> (E.z z = A <-> E.x x = A))
12	11	3adant3 975	. . . 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 201	. . 3 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> E.x x = A)
14		bi1 178	. . . . . . . . 9 |- ((ph <-> ps) -> (ph -> ps))
15	14	imim2i 13	. . . . . . . 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 418	. . . . . 6 |- (((x = A -> (ph <-> ps)) /\ ph) -> (x = A -> ps))
18	17	alanimi 1562	. . . . 5 |- ((A.x(x = A -> (ph <-> ps)) /\ A.xph) -> A.x(x = A -> ps))
19	18	3ad2ant2 977	. . . 4 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> A.x(x = A -> ps))
20		simp1r 980	. . . . 5 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> F/xps)
21		19.23t 1800	. . . . 5 |- ( F/xps -> (A.x(x = A -> ps) <-> (E.x x = A -> ps)))
22	20, 21	syl 15	. . . 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 201	. . 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 14	. 2 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. _V) -> ps)
25	1, 24	syl3an3 1217	1 |- ((( F/_xA /\ F/xps) /\ (A.x(x = A -> (ph <-> ps)) /\ A.xph) /\ A e. V) -> ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710   F/_wnfc 2477  _Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  vtocldf  2907  iota2df  4366