Theorem vtoclgft 2776

Description: Closed theorem form of vtoclgf 2784. (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 2737	. 2 |- ( A e. V -> A e. _V )
2		elisset 2740	. . . . 5 |- ( A e. _V -> E. z z = A )
3	2	3ad2ant3 1010	. . . 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 2311	. . . . . . 7 |- F/ x F/_ x A
5		nfcvd 2309	. . . . . . . 8 |- ( F/_ x A -> F/_ x z )
6		id 19	. . . . . . . 8 |- ( F/_ x A -> F/_ x A )
7	5, 6	nfeqd 2323	. . . . . . 7 |- ( F/_ x A -> F/ x z = A )
8		eqeq1 2172	. . . . . . . 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 1915	. . . . . 6 |- ( F/_ x A -> ( E. z z = A <-> E. x x = A ) )
11	10	ad2antrr 480	. . . . 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 1007	. . . 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 146	. . 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 117	. . . . . . . . 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 123	. . . . . 6 |- ( ( ( x = A -> ( ph <-> ps ) ) /\ ph ) -> ( x = A -> ps ) )
18	17	alanimi 1447	. . . . 5 |- ( ( A. x ( x = A -> ( ph <-> ps ) ) /\ A. x ph ) -> A. x ( x = A -> ps ) )
19	18	3ad2ant2 1009	. . . 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 1012	. . . . 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 1665	. . . . 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 146	. . 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 1263	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 103   <-> wb 104   /\ w3a 968   A.wal 1341   = wceq 1343   F/wnf 1448   E.wex 1480   e. wcel 2136   F/_wnfc 2295   _Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  vtocldf  2777