Theorem vtoclgft 3157

Description: Closed theorem form of vtoclgf 3165. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 12-Oct-2016.)

Assertion

Ref	Expression
vtoclgft

Proof of Theorem vtoclgft

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3118	. 2
2		elisset 3120	. . . . 5
3	2	3ad2ant3 1019	. . . 4
4		nfnfc1 2622	. . . . . . 7
5		nfcvd 2620	. . . . . . . 8
6		id 22	. . . . . . . 8
7	5, 6	nfeqd 2626	. . . . . . 7
8		eqeq1 2461	. . . . . . . 8
9	8	a1i 11	. . . . . . 7
10	4, 7, 9	cbvexd 2026	. . . . . 6
11	10	ad2antrr 725	. . . . 5
12	11	3adant3 1016	. . . 4
13	3, 12	mpbid 210	. . 3
14		bi1 186	. . . . . . . . 9
15	14	imim2i 14	. . . . . . . 8
16	15	com23 78	. . . . . . 7
17	16	imp 429	. . . . . 6
18	17	alanimi 1637	. . . . 5
19	18	3ad2ant2 1018	. . . 4
20		simp1r 1021	. . . . 5
21		19.23t 1909	. . . . 5
22	20, 21	syl 16	. . . 4
23	19, 22	mpbid 210	. . 3
24	13, 23	mpd 15	. 2
25	1, 24	syl3an3 1263	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  A.wal 1393  =wceq 1395  E.wex 1612  F/wnf 1616  e.wcel 1818  F/_wnfc 2605  cvv 3109

This theorem is referenced by:  vtocldf  3158  bj-vtoclgfALT  34588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111