MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ceqsex8v Unicode version

Theorem ceqsex8v 2829
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 19.42vv 1848 . . . . . . 7  |-  ( 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. t E. s ( ( v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ph )
) )
212exbii 1570 . . . . . 6  |-  ( 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 ( ( ( x  =  A  /\  y  =  B )  /\  ( z  =  C  /\  w  =  D ) )  /\  E. t E. s ( ( v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ph )
) )
3 19.42vv 1848 . . . . . 6  |-  ( E. v E. u ( ( ( x  =  A  /\  y  =  B )  /\  (
z  =  C  /\  w  =  D )
)  /\  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 )
) )
42, 3bitri 240 . . . . 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 )
) )
5 3anass 938 . . . . . . . 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 ) ) )
6 df-3an 936 . . . . . . . . 9  |-  ( ( ( v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ph ) 
<->  ( ( ( v  =  E  /\  u  =  F )  /\  (
t  =  G  /\  s  =  H )
)  /\  ph ) )
76anbi2i 675 . . . . . . . 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 ) ) )
85, 7bitr4i 243 . . . . . . 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 )
) )
982exbii 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 )
) )
1092exbii 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 ) ) )
11 df-3an 936 . . . . 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 )
) )
124, 10, 113bitr4i 268 . . . 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 ) ) )
13122exbii 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 ) ) )
14132exbii 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 ) ) )
15 ceqsex8v.1 . . . 4  |-  A  e. 
_V
16 ceqsex8v.2 . . . 4  |-  B  e. 
_V
17 ceqsex8v.3 . . . 4  |-  C  e. 
_V
18 ceqsex8v.4 . . . 4  |-  D  e. 
_V
19 ceqsex8v.9 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
20193anbi3d 1258 . . . . 5  |-  ( x  =  A  ->  (
( ( v  =  E  /\  u  =  F )  /\  (
t  =  G  /\  s  =  H )  /\  ph )  <->  ( (
v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ps )
) )
21204exbidv 1616 . . . 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 )
) )
22 ceqsex8v.10 . . . . . 6  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
23223anbi3d 1258 . . . . 5  |-  ( y  =  B  ->  (
( ( v  =  E  /\  u  =  F )  /\  (
t  =  G  /\  s  =  H )  /\  ps )  <->  ( (
v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ch )
) )
24234exbidv 1616 . . . 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 )
) )
25 ceqsex8v.11 . . . . . 6  |-  ( z  =  C  ->  ( ch 
<->  th ) )
26253anbi3d 1258 . . . . 5  |-  ( z  =  C  ->  (
( ( v  =  E  /\  u  =  F )  /\  (
t  =  G  /\  s  =  H )  /\  ch )  <->  ( (
v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  th )
) )
27264exbidv 1616 . . . 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 )
) )
28 ceqsex8v.12 . . . . . 6  |-  ( w  =  D  ->  ( th 
<->  ta ) )
29283anbi3d 1258 . . . . 5  |-  ( w  =  D  ->  (
( ( v  =  E  /\  u  =  F )  /\  (
t  =  G  /\  s  =  H )  /\  th )  <->  ( (
v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ta )
) )
30294exbidv 1616 . . . 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 )
) )
3115, 16, 17, 18, 21, 24, 27, 30ceqsex4v 2827 . . 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 ) )
32 ceqsex8v.5 . . . 4  |-  E  e. 
_V
33 ceqsex8v.6 . . . 4  |-  F  e. 
_V
34 ceqsex8v.7 . . . 4  |-  G  e. 
_V
35 ceqsex8v.8 . . . 4  |-  H  e. 
_V
36 ceqsex8v.13 . . . 4  |-  ( v  =  E  ->  ( ta 
<->  et ) )
37 ceqsex8v.14 . . . 4  |-  ( u  =  F  ->  ( et 
<->  ze ) )
38 ceqsex8v.15 . . . 4  |-  ( t  =  G  ->  ( ze 
<-> 
si ) )
39 ceqsex8v.16 . . . 4  |-  ( s  =  H  ->  ( si 
<->  rh ) )
4032, 33, 34, 35, 36, 37, 38, 39ceqsex4v 2827 . . 3  |-  ( E. v E. u E. t E. s ( ( v  =  E  /\  u  =  F )  /\  ( t  =  G  /\  s  =  H )  /\  ta )  <->  rh )
4131, 40bitri 240 . 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 )
4214, 41bitri 240 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    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790
  Copyright terms: Public domain W3C validator