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Theorem copsexgb 24966
Description: Substitution of class  A for ordered triple  <. <. x ,  y >. ,  z
>.. See copsexg 4252. (Contributed by FL, 10-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
copsexgb  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  <->  E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) ) )
Distinct variable group:    x, y, z, A
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem copsexgb
Dummy variables  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.8a 1718 . . . 4  |-  ( ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
2 19.8a 1718 . . . 4  |-  ( E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
3 19.8a 1718 . . . 4  |-  ( E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
41, 2, 33syl 18 . . 3  |-  ( ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
54ex 423 . 2  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  ->  E. x E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph ) ) )
6 vex 2791 . . . 4  |-  x  e. 
_V
7 vex 2791 . . . 4  |-  y  e. 
_V
8 vex 2791 . . . 4  |-  z  e. 
_V
96, 7, 8eqvinopb 24965 . . 3  |-  ( A  =  <. <. x ,  y
>. ,  z >.  <->  E. u E. v E. w
( A  =  <. <.
u ,  v >. ,  w >.  /\  <. <. u ,  v >. ,  w >.  =  <. <. x ,  y
>. ,  z >. ) )
10 eqeq1 2289 . . . . . . . . . . . . . . 15  |-  ( A  =  <. <. u ,  v
>. ,  w >.  -> 
( A  =  <. <.
x ,  y >. ,  z >.  <->  <. <. u ,  v >. ,  w >.  =  <. <. x ,  y
>. ,  z >. ) )
11 eqcom 2285 . . . . . . . . . . . . . . . . 17  |-  ( <. <. u ,  v >. ,  w >.  =  <. <.
x ,  y >. ,  z >.  <->  <. <. x ,  y >. ,  z
>.  =  <. <. u ,  v >. ,  w >. )
126, 7, 8otth2 4249 . . . . . . . . . . . . . . . . 17  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
u ,  v >. ,  w >.  <->  ( x  =  u  /\  y  =  v  /\  z  =  w ) )
1311, 12bitri 240 . . . . . . . . . . . . . . . 16  |-  ( <. <. u ,  v >. ,  w >.  =  <. <.
x ,  y >. ,  z >.  <->  ( x  =  u  /\  y  =  v  /\  z  =  w ) )
14 df-3an 936 . . . . . . . . . . . . . . . 16  |-  ( ( x  =  u  /\  y  =  v  /\  z  =  w )  <->  ( ( x  =  u  /\  y  =  v )  /\  z  =  w ) )
1513, 14bitri 240 . . . . . . . . . . . . . . 15  |-  ( <. <. u ,  v >. ,  w >.  =  <. <.
x ,  y >. ,  z >.  <->  ( (
x  =  u  /\  y  =  v )  /\  z  =  w
) )
1610, 15syl6bb 252 . . . . . . . . . . . . . 14  |-  ( A  =  <. <. u ,  v
>. ,  w >.  -> 
( A  =  <. <.
x ,  y >. ,  z >.  <->  ( (
x  =  u  /\  y  =  v )  /\  z  =  w
) ) )
1716anbi1d 685 . . . . . . . . . . . . 13  |-  ( A  =  <. <. u ,  v
>. ,  w >.  -> 
( ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ( (
( x  =  u  /\  y  =  v )  /\  z  =  w )  /\  ph ) ) )
18 anass 630 . . . . . . . . . . . . 13  |-  ( ( ( ( x  =  u  /\  y  =  v )  /\  z  =  w )  /\  ph ) 
<->  ( ( x  =  u  /\  y  =  v )  /\  (
z  =  w  /\  ph ) ) )
1917, 18syl6bb 252 . . . . . . . . . . . 12  |-  ( A  =  <. <. u ,  v
>. ,  w >.  -> 
( ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ( (
x  =  u  /\  y  =  v )  /\  ( z  =  w  /\  ph ) ) ) )
2019exbidv 1612 . . . . . . . . . . 11  |-  ( A  =  <. <. u ,  v
>. ,  w >.  -> 
( E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  E. z
( ( x  =  u  /\  y  =  v )  /\  (
z  =  w  /\  ph ) ) ) )
21 19.42v 1846 . . . . . . . . . . . 12  |-  ( E. z ( ( x  =  u  /\  y  =  v )  /\  ( z  =  w  /\  ph ) )  <-> 
( ( x  =  u  /\  y  =  v )  /\  E. z ( z  =  w  /\  ph )
) )
22 anass 630 . . . . . . . . . . . 12  |-  ( ( ( x  =  u  /\  y  =  v )  /\  E. z
( z  =  w  /\  ph ) )  <-> 
( x  =  u  /\  ( y  =  v  /\  E. z
( z  =  w  /\  ph ) ) ) )
2321, 22bitri 240 . . . . . . . . . . 11  |-  ( E. z ( ( x  =  u  /\  y  =  v )  /\  ( z  =  w  /\  ph ) )  <-> 
( x  =  u  /\  ( y  =  v  /\  E. z
( z  =  w  /\  ph ) ) ) )
2420, 23syl6bb 252 . . . . . . . . . 10  |-  ( A  =  <. <. u ,  v
>. ,  w >.  -> 
( E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  ( x  =  u  /\  (
y  =  v  /\  E. z ( z  =  w  /\  ph )
) ) ) )
2524exbidv 1612 . . . . . . . . 9  |-  ( A  =  <. <. u ,  v
>. ,  w >.  -> 
( E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. y
( x  =  u  /\  ( y  =  v  /\  E. z
( z  =  w  /\  ph ) ) ) ) )
26 19.42v 1846 . . . . . . . . 9  |-  ( E. y ( x  =  u  /\  ( y  =  v  /\  E. z ( z  =  w  /\  ph )
) )  <->  ( x  =  u  /\  E. y
( y  =  v  /\  E. z ( z  =  w  /\  ph ) ) ) )
2725, 26syl6bb 252 . . . . . . . 8  |-  ( A  =  <. <. u ,  v
>. ,  w >.  -> 
( E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ( x  =  u  /\  E. y
( y  =  v  /\  E. z ( z  =  w  /\  ph ) ) ) ) )
2827exbidv 1612 . . . . . . 7  |-  ( A  =  <. <. u ,  v
>. ,  w >.  -> 
( E. x E. y E. z ( A  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x
( x  =  u  /\  E. y ( y  =  v  /\  E. z ( z  =  w  /\  ph )
) ) ) )
2928adantr 451 . . . . . 6  |-  ( ( A  =  <. <. u ,  v >. ,  w >.  /\  <. <. u ,  v
>. ,  w >.  = 
<. <. x ,  y
>. ,  z >. )  ->  ( E. x E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  <->  E. x
( x  =  u  /\  E. y ( y  =  v  /\  E. z ( z  =  w  /\  ph )
) ) ) )
30 simpr1 961 . . . . . . . . 9  |-  ( ( A  =  <. <. u ,  v >. ,  w >.  /\  ( x  =  u  /\  y  =  v  /\  z  =  w ) )  ->  x  =  u )
31 euequ1 2231 . . . . . . . . . 10  |-  E! x  x  =  u
32 eupick 2206 . . . . . . . . . 10  |-  ( ( E! x  x  =  u  /\  E. x
( x  =  u  /\  E. y ( y  =  v  /\  E. z ( z  =  w  /\  ph )
) ) )  -> 
( x  =  u  ->  E. y ( y  =  v  /\  E. z ( z  =  w  /\  ph )
) ) )
3331, 32mpan 651 . . . . . . . . 9  |-  ( E. x ( x  =  u  /\  E. y
( y  =  v  /\  E. z ( z  =  w  /\  ph ) ) )  -> 
( x  =  u  ->  E. y ( y  =  v  /\  E. z ( z  =  w  /\  ph )
) ) )
3430, 33syl5com 26 . . . . . . . 8  |-  ( ( A  =  <. <. u ,  v >. ,  w >.  /\  ( x  =  u  /\  y  =  v  /\  z  =  w ) )  -> 
( E. x ( x  =  u  /\  E. y ( y  =  v  /\  E. z
( z  =  w  /\  ph ) ) )  ->  E. y
( y  =  v  /\  E. z ( z  =  w  /\  ph ) ) ) )
35 simpr2 962 . . . . . . . . 9  |-  ( ( A  =  <. <. u ,  v >. ,  w >.  /\  ( x  =  u  /\  y  =  v  /\  z  =  w ) )  -> 
y  =  v )
36 euequ1 2231 . . . . . . . . . 10  |-  E! y  y  =  v
37 eupick 2206 . . . . . . . . . 10  |-  ( ( E! y  y  =  v  /\  E. y
( y  =  v  /\  E. z ( z  =  w  /\  ph ) ) )  -> 
( y  =  v  ->  E. z ( z  =  w  /\  ph ) ) )
3836, 37mpan 651 . . . . . . . . 9  |-  ( E. y ( y  =  v  /\  E. z
( z  =  w  /\  ph ) )  ->  ( y  =  v  ->  E. z
( z  =  w  /\  ph ) ) )
3935, 38syl5com 26 . . . . . . . 8  |-  ( ( A  =  <. <. u ,  v >. ,  w >.  /\  ( x  =  u  /\  y  =  v  /\  z  =  w ) )  -> 
( E. y ( y  =  v  /\  E. z ( z  =  w  /\  ph )
)  ->  E. z
( z  =  w  /\  ph ) ) )
40 simpr3 963 . . . . . . . . 9  |-  ( ( A  =  <. <. u ,  v >. ,  w >.  /\  ( x  =  u  /\  y  =  v  /\  z  =  w ) )  -> 
z  =  w )
41 euequ1 2231 . . . . . . . . . 10  |-  E! z  z  =  w
42 eupick 2206 . . . . . . . . . 10  |-  ( ( E! z  z  =  w  /\  E. z
( z  =  w  /\  ph ) )  ->  ( z  =  w  ->  ph ) )
4341, 42mpan 651 . . . . . . . . 9  |-  ( E. z ( z  =  w  /\  ph )  ->  ( z  =  w  ->  ph ) )
4440, 43syl5com 26 . . . . . . . 8  |-  ( ( A  =  <. <. u ,  v >. ,  w >.  /\  ( x  =  u  /\  y  =  v  /\  z  =  w ) )  -> 
( E. z ( z  =  w  /\  ph )  ->  ph ) )
4534, 39, 443syld 51 . . . . . . 7  |-  ( ( A  =  <. <. u ,  v >. ,  w >.  /\  ( x  =  u  /\  y  =  v  /\  z  =  w ) )  -> 
( E. x ( x  =  u  /\  E. y ( y  =  v  /\  E. z
( z  =  w  /\  ph ) ) )  ->  ph ) )
4613, 45sylan2b 461 . . . . . 6  |-  ( ( A  =  <. <. u ,  v >. ,  w >.  /\  <. <. u ,  v
>. ,  w >.  = 
<. <. x ,  y
>. ,  z >. )  ->  ( E. x
( x  =  u  /\  E. y ( y  =  v  /\  E. z ( z  =  w  /\  ph )
) )  ->  ph )
)
4729, 46sylbid 206 . . . . 5  |-  ( ( A  =  <. <. u ,  v >. ,  w >.  /\  <. <. u ,  v
>. ,  w >.  = 
<. <. x ,  y
>. ,  z >. )  ->  ( E. x E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  ph ) )
4847exlimiv 1666 . . . 4  |-  ( E. w ( A  = 
<. <. u ,  v
>. ,  w >.  /\ 
<. <. u ,  v
>. ,  w >.  = 
<. <. x ,  y
>. ,  z >. )  ->  ( E. x E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  ph ) )
4948exlimivv 1667 . . 3  |-  ( E. u E. v E. w ( A  = 
<. <. u ,  v
>. ,  w >.  /\ 
<. <. u ,  v
>. ,  w >.  = 
<. <. x ,  y
>. ,  z >. )  ->  ( E. x E. y E. z ( A  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  ph ) )
509, 49sylbi 187 . 2  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( A  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
)
515, 50impbid 183 1  |-  ( A  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  <->  E. x E. y E. z ( A  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623   E!weu 2143   <.cop 3643
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649
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