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Theorem ceqsex8v 2705
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 1867 . . . . 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 951 . . . . . . . 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 949 . . . . . . . . 9  |-  ( ( ( v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ph ) 
<->  ( ( ( v  =  E  /\  u  =  F )  /\  (
t  =  G  /\  s  =  H )
)  /\  ph ) )
43anbi2i 452 . . . . . . . 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 186 . . . . . . 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 1570 . . . . . 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 1570 . . . . 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 949 . . . . 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 211 . . . 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 1570 . . 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 1570 . 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 1281 . . . . 5  |-  ( x  =  A  ->  (
( ( v  =  E  /\  u  =  F )  /\  (
t  =  G  /\  s  =  H )  /\  ph )  <->  ( (
v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ps )
) )
18174exbidv 1826 . . . 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 1281 . . . . 5  |-  ( y  =  B  ->  (
( ( v  =  E  /\  u  =  F )  /\  (
t  =  G  /\  s  =  H )  /\  ps )  <->  ( (
v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ch )
) )
21204exbidv 1826 . . . 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 1281 . . . . 5  |-  ( z  =  C  ->  (
( ( v  =  E  /\  u  =  F )  /\  (
t  =  G  /\  s  =  H )  /\  ch )  <->  ( (
v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  th )
) )
24234exbidv 1826 . . . 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 1281 . . . . 5  |-  ( w  =  D  ->  (
( ( v  =  E  /\  u  =  F )  /\  (
t  =  G  /\  s  =  H )  /\  th )  <->  ( (
v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ta )
) )
27264exbidv 1826 . . . 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 2703 . . 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 2703 . . 3  |-  ( E. v E. u E. t E. s ( ( v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ta )  <->  rh )
3828, 37bitri 183 . 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 183 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 103    <-> wb 104    /\ w3a 947    = wceq 1316   E.wex 1453    e. wcel 1465   _Vcvv 2660
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by: (None)
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