Theorem ceqsex8v 2593

Description: Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Hypotheses

Ref	Expression
ceqsex8v.1	|- A e. _V
ceqsex8v.2	|- B e. _V
ceqsex8v.3	|- C e. _V
ceqsex8v.4	|- D e. _V
ceqsex8v.5	|- E e. _V
ceqsex8v.6	|- F e. _V
ceqsex8v.7	|- G e. _V
ceqsex8v.8	|- H e. _V
ceqsex8v.9	|- (x = A -> (ph <-> ps))
ceqsex8v.10	|- (y = B -> (ps <-> ch))
ceqsex8v.11	|- (z = C -> (ch <-> th))
ceqsex8v.12	|- (w = D -> (th <-> ta))
ceqsex8v.13	|- (v = E -> (ta <-> et))
ceqsex8v.14	|- (u = F -> (et <-> ze))
ceqsex8v.15	|- ( t = G -> (ze <-> si))
ceqsex8v.16	|- ( s = H -> (si <-> rh))

Assertion

Ref	Expression
ceqsex8v	|- (E.xE.yE.zE.wE.vE.uE. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph) <-> rh)

Distinct variable groups:   x,y,z,w,v,u, t, s,A   x,B,y,z,w,v,u, t, s   x, C,y,z,w,v,u, t, s   x, D,y,z,w,v,u, t, s   x, E,y,z,w,v,u, t, s   x, F,y,z,w,v,u, t, s   x, G,y,z,w,v,u, t, s   x, H,y,z,w,v,u, t, s   ps,x   ch,y   th,z   ta,w   et,v   ze,u   si, t   rh, s

Allowed substitution hints:   ph(x,y,z,w,v,u, t, s)   ps(y,z,w,v,u, t, s)   ch(x,z,w,v,u, t, s)   th(x,y,w,v,u, t, s)   ta(x,y,z,v,u, t, s)   et(x,y,z,w,u, t, s)   ze(x,y,z,w,v, t, s)   si(x,y,z,w,v,u, s)   rh(x,y,z,w,v,u, t)

Proof of Theorem ceqsex8v

Step	Hyp	Ref	Expression
1		19.42vvvv 1787	. . . . 5 |- (E.vE.uE. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)) <-> (((x = A /\ y = B) /\ (z = C /\ w = D)) /\ E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)))
2		3anass 888	. . . . . . . 8 |- ((((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph) <-> (((x = A /\ y = B) /\ (z = C /\ w = D)) /\ (((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph)))
3		df-3an 886	. . . . . . . . 9 |- (((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph) <-> (((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph))
4	3	anbi2i 430	. . . . . . . 8 |- ((((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)) <-> (((x = A /\ y = B) /\ (z = C /\ w = D)) /\ (((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph)))
5	2, 4	bitr4i 176	. . . . . . 7 |- ((((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph) <-> (((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)))
6	5	2exbii 1494	. . . . . 6 |- (E. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph) <-> E. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)))
7	6	2exbii 1494	. . . . 5 |- (E.vE.uE. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph) <-> E.vE.uE. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)))
8		df-3an 886	. . . . 5 |- (((x = A /\ y = B) /\ (z = C /\ w = D) /\ E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)) <-> (((x = A /\ y = B) /\ (z = C /\ w = D)) /\ E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)))
9	1, 7, 8	3bitr4i 201	. . . 4 |- (E.vE.uE. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph) <-> ((x = A /\ y = B) /\ (z = C /\ w = D) /\ E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)))
10	9	2exbii 1494	. . 3 |- (E.zE.wE.vE.uE. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph) <-> E.zE.w((x = A /\ y = B) /\ (z = C /\ w = D) /\ E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)))
11	10	2exbii 1494	. 2 |- (E.xE.yE.zE.wE.vE.uE. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph) <-> E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D) /\ E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)))
12		ceqsex8v.1	. . . 4 |- A e. _V
13		ceqsex8v.2	. . . 4 |- B e. _V
14		ceqsex8v.3	. . . 4 |- C e. _V
15		ceqsex8v.4	. . . 4 |- D e. _V
16		ceqsex8v.9	. . . . . 6 |- (x = A -> (ph <-> ps))
17	16	3anbi3d 1212	. . . . 5 |- (x = A -> (((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph) <-> ((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ps)))
18	17	4exbidv 1747	. . . 4 |- (x = A -> (E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph) <-> E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ps)))
19		ceqsex8v.10	. . . . . 6 |- (y = B -> (ps <-> ch))
20	19	3anbi3d 1212	. . . . 5 |- (y = B -> (((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ps) <-> ((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ch)))
21	20	4exbidv 1747	. . . 4 |- (y = B -> (E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ps) <-> E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ch)))
22		ceqsex8v.11	. . . . . 6 |- (z = C -> (ch <-> th))
23	22	3anbi3d 1212	. . . . 5 |- (z = C -> (((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ch) <-> ((v = E /\ u = F) /\ ( t = G /\ s = H) /\ th)))
24	23	4exbidv 1747	. . . 4 |- (z = C -> (E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ch) <-> E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ th)))
25		ceqsex8v.12	. . . . . 6 |- (w = D -> (th <-> ta))
26	25	3anbi3d 1212	. . . . 5 |- (w = D -> (((v = E /\ u = F) /\ ( t = G /\ s = H) /\ th) <-> ((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ta)))
27	26	4exbidv 1747	. . . 4 |- (w = D -> (E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ th) <-> E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ta)))
28	12, 13, 14, 15, 18, 21, 24, 27	ceqsex4v 2591	. . 3 |- (E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D) /\ E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)) <-> E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ta))
29		ceqsex8v.5	. . . 4 |- E e. _V
30		ceqsex8v.6	. . . 4 |- F e. _V
31		ceqsex8v.7	. . . 4 |- G e. _V
32		ceqsex8v.8	. . . 4 |- H e. _V
33		ceqsex8v.13	. . . 4 |- (v = E -> (ta <-> et))
34		ceqsex8v.14	. . . 4 |- (u = F -> (et <-> ze))
35		ceqsex8v.15	. . . 4 |- ( t = G -> (ze <-> si))
36		ceqsex8v.16	. . . 4 |- ( s = H -> (si <-> rh))
37	29, 30, 31, 32, 33, 34, 35, 36	ceqsex4v 2591	. . 3 |- (E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ta) <-> rh)
38	28, 37	bitri 173	. 2 |- (E.xE.yE.zE.w((x = A /\ y = B) /\ (z = C /\ w = D) /\ E.vE.uE. tE. s((v = E /\ u = F) /\ ( t = G /\ s = H) /\ ph)) <-> rh)
39	11, 38	bitri 173	1 |- (E.xE.yE.zE.wE.vE.uE. tE. s(((x = A /\ y = B) /\ (z = C /\ w = D)) /\ ((v = E /\ u = F) /\ ( t = G /\ s = H)) /\ ph) <-> rh)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 884   = wceq 1242  E.wex 1378   e. wcel 1390   _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-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-v 2553

This theorem is referenced by: (None)