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Theorem ceqsex8v 2599
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. x E. y E. z E. w E. v E. u E. t E. 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
StepHypRef Expression
1 19.42vvvv 1790 . . . . 5 |- ( E. v E. u E. t E. 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. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) ) )
2 3anass 889 . . . . . . . 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 887 . . . . . . . . 9 |- ( ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) <-> ( ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) )
43anbi2i 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 ) ) )
52, 4bitr4i 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 ) ) )
652exbii 1497 . . . . . 6 |- ( E. t E. s ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) <-> E. t E. s ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) ) )
762exbii 1497 . . . . 5 |- ( E. v E. u E. t E. s ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) <-> E. v E. u E. t E. s ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) ) )
8 df-3an 887 . . . . 5 |- ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) ) <-> ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) ) )
91, 7, 83bitr4i 201 . . . 4 |- ( E. v E. u E. t E. 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. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) ) )
1092exbii 1497 . . 3 |- ( E. z E. w E. v E. u E. t E. s ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) <-> E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) ) )
11102exbii 1497 . 2 |- ( E. x E. y E. z E. w E. v E. u E. t E. s ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) <-> E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ E. v E. u E. t E. 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 ) )
17163anbi3d 1213 . . . . 5 |- ( x = A -> ( ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) <-> ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ps ) ) )
18174exbidv 1750 . . . 4 |- ( x = A -> ( E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) <-> E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ps ) ) )
19 ceqsex8v.10 . . . . . 6 |- ( y = B -> ( ps <-> ch ) )
20193anbi3d 1213 . . . . 5 |- ( y = B -> ( ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ps ) <-> ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ch ) ) )
21204exbidv 1750 . . . 4 |- ( y = B -> ( E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ps ) <-> E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ch ) ) )
22 ceqsex8v.11 . . . . . 6 |- ( z = C -> ( ch <-> th ) )
23223anbi3d 1213 . . . . 5 |- ( z = C -> ( ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ch ) <-> ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ th ) ) )
24234exbidv 1750 . . . 4 |- ( z = C -> ( E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ch ) <-> E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ th ) ) )
25 ceqsex8v.12 . . . . . 6 |- ( w = D -> ( th <-> ta ) )
26253anbi3d 1213 . . . . 5 |- ( w = D -> ( ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ th ) <-> ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ta ) ) )
27264exbidv 1750 . . . 4 |- ( w = D -> ( E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ th ) <-> E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ta ) ) )
2812, 13, 14, 15, 18, 21, 24, 27ceqsex4v 2597 . . 3 |- ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) ) <-> E. v E. u E. t E. 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 ) )
3729, 30, 31, 32, 33, 34, 35, 36ceqsex4v 2597 . . 3 |- ( E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ta ) <-> rh )
3828, 37bitri 173 . 2 |- ( E. x E. y E. z E. w ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) /\ E. v E. u E. t E. s ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) /\ ph ) ) <-> rh )
3911, 38bitri 173 1 |- ( E. x E. y E. z E. w E. v E. u E. t E. s ( ( ( x = A /\ y = B ) /\ ( z = C /\ w = D ) ) /\ ( ( v = E /\ u = F ) /\ ( t = G /\ s = H ) ) /\ ph ) <-> rh )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557
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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559
This theorem is referenced by: (None)
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