MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ceqsex6v Structured version   Unicode version

Theorem ceqsex6v 2997
Description: Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
Hypotheses
Ref Expression
ceqsex6v.1  |-  A  e. 
_V
ceqsex6v.2  |-  B  e. 
_V
ceqsex6v.3  |-  C  e. 
_V
ceqsex6v.4  |-  D  e. 
_V
ceqsex6v.5  |-  E  e. 
_V
ceqsex6v.6  |-  F  e. 
_V
ceqsex6v.7  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
ceqsex6v.8  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
ceqsex6v.9  |-  ( z  =  C  ->  ( ch 
<->  th ) )
ceqsex6v.10  |-  ( w  =  D  ->  ( th 
<->  ta ) )
ceqsex6v.11  |-  ( v  =  E  ->  ( ta 
<->  et ) )
ceqsex6v.12  |-  ( u  =  F  ->  ( et 
<->  ze ) )
Assertion
Ref Expression
ceqsex6v  |-  ( E. x E. y E. z E. w E. v E. u ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph )  <->  ze )
Distinct variable groups:    x, y,
z, w, v, u, A    x, B, y, z, w, v, u   
x, C, y, z, w, v, u    x, D, y, z, w, v, u    x, E, y, z, w, v, u   
x, F, y, z, w, v, u    ps, x    ch, y    th, z    ta, w    et, v    ze, u
Allowed substitution hints:    ph( x, y, z, w, v, u)    ps( y, z, w, v, u)    ch( x, z, w, v, u)    th( x, y, w, v, u)    ta( x, y, z, v, u)    et( x, y, z, w, u)    ze( x, y, z, w, v)

Proof of Theorem ceqsex6v
StepHypRef Expression
1 3anass 941 . . . . 5  |-  ( ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph ) 
<->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  (
( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph )
) )
213exbii 1595 . . . 4  |-  ( E. w E. v E. u ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  (
w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph )  <->  E. w E. v E. u ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph ) ) )
3 19.42vvv 1932 . . . 4  |-  ( E. w E. v E. u ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  (
( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph )
)  <->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  E. w E. v E. u
( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph ) ) )
42, 3bitri 242 . . 3  |-  ( E. w E. v E. u ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  (
w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph )  <->  ( (
x  =  A  /\  y  =  B  /\  z  =  C )  /\  E. w E. v E. u ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph ) ) )
543exbii 1595 . 2  |-  ( E. x E. y E. z E. w E. v E. u ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph )  <->  E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  E. w E. v E. u
( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph ) ) )
6 ceqsex6v.1 . . . 4  |-  A  e. 
_V
7 ceqsex6v.2 . . . 4  |-  B  e. 
_V
8 ceqsex6v.3 . . . 4  |-  C  e. 
_V
9 ceqsex6v.7 . . . . . 6  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
109anbi2d 686 . . . . 5  |-  ( x  =  A  ->  (
( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph ) 
<->  ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ps ) ) )
11103exbidv 1640 . . . 4  |-  ( x  =  A  ->  ( E. w E. v E. u ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph ) 
<->  E. w E. v E. u ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ps ) ) )
12 ceqsex6v.8 . . . . . 6  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
1312anbi2d 686 . . . . 5  |-  ( y  =  B  ->  (
( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ps ) 
<->  ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ch ) ) )
14133exbidv 1640 . . . 4  |-  ( y  =  B  ->  ( E. w E. v E. u ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ps ) 
<->  E. w E. v E. u ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ch ) ) )
15 ceqsex6v.9 . . . . . 6  |-  ( z  =  C  ->  ( ch 
<->  th ) )
1615anbi2d 686 . . . . 5  |-  ( z  =  C  ->  (
( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ch ) 
<->  ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  th ) ) )
17163exbidv 1640 . . . 4  |-  ( z  =  C  ->  ( E. w E. v E. u ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ch ) 
<->  E. w E. v E. u ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  th ) ) )
186, 7, 8, 11, 14, 17ceqsex3v 2995 . . 3  |-  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  E. w E. v E. u
( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph ) )  <->  E. w E. v E. u ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  th )
)
19 ceqsex6v.4 . . . 4  |-  D  e. 
_V
20 ceqsex6v.5 . . . 4  |-  E  e. 
_V
21 ceqsex6v.6 . . . 4  |-  F  e. 
_V
22 ceqsex6v.10 . . . 4  |-  ( w  =  D  ->  ( th 
<->  ta ) )
23 ceqsex6v.11 . . . 4  |-  ( v  =  E  ->  ( ta 
<->  et ) )
24 ceqsex6v.12 . . . 4  |-  ( u  =  F  ->  ( et 
<->  ze ) )
2519, 20, 21, 22, 23, 24ceqsex3v 2995 . . 3  |-  ( E. w E. v E. u ( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  th ) 
<->  ze )
2618, 25bitri 242 . 2  |-  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  E. w E. v E. u
( ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph ) )  <->  ze )
275, 26bitri 242 1  |-  ( E. x E. y E. z E. w E. v E. u ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ( w  =  D  /\  v  =  E  /\  u  =  F )  /\  ph )  <->  ze )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-an 362  df-3an 939  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-v 2959
  Copyright terms: Public domain W3C validator