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Theorem oprabid 5568
Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. Although this theorem would be useful with a distinct variable constraint between  x,  y, and  z, we use ax-bndl 1440 to eliminate that constraint. (Contributed by Mario Carneiro, 20-Mar-2013.)
Assertion
Ref Expression
oprabid  |-  ( <. <. x ,  y >. ,  z >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ph )

Proof of Theorem oprabid
Dummy variables  a  r  s  t  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2605 . . . 4  |-  x  e. 
_V
2 vex 2605 . . . 4  |-  y  e. 
_V
31, 2opex 3992 . . 3  |-  <. x ,  y >.  e.  _V
4 vex 2605 . . 3  |-  z  e. 
_V
5 opexg 3991 . . 3  |-  ( (
<. x ,  y >.  e.  _V  /\  z  e. 
_V )  ->  <. <. x ,  y >. ,  z
>.  e.  _V )
63, 4, 5mp2an 417 . 2  |-  <. <. x ,  y >. ,  z
>.  e.  _V
73, 4eqvinop 4006 . . . . 5  |-  ( w  =  <. <. x ,  y
>. ,  z >.  <->  E. a E. t ( w  =  <. a ,  t
>.  /\  <. a ,  t
>.  =  <. <. x ,  y >. ,  z
>. ) )
87biimpi 118 . . . 4  |-  ( w  =  <. <. x ,  y
>. ,  z >.  ->  E. a E. t ( w  =  <. a ,  t >.  /\  <. a ,  t >.  =  <. <.
x ,  y >. ,  z >. )
)
9 eqeq1 2088 . . . . . . . 8  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  <->  <. a ,  t >.  =  <. <.
x ,  y >. ,  z >. )
)
10 vex 2605 . . . . . . . . 9  |-  a  e. 
_V
11 vex 2605 . . . . . . . . 9  |-  t  e. 
_V
1210, 11opth1 3999 . . . . . . . 8  |-  ( <.
a ,  t >.  =  <. <. x ,  y
>. ,  z >.  -> 
a  =  <. x ,  y >. )
139, 12syl6bi 161 . . . . . . 7  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
a  =  <. x ,  y >. )
)
141, 2eqvinop 4006 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>. 
<->  E. r E. s
( a  =  <. r ,  s >.  /\  <. r ,  s >.  =  <. x ,  y >. )
)
15 opeq1 3578 . . . . . . . . . . . . 13  |-  ( a  =  <. r ,  s
>.  ->  <. a ,  t
>.  =  <. <. r ,  s >. ,  t
>. )
1615eqeq2d 2093 . . . . . . . . . . . 12  |-  ( a  =  <. r ,  s
>.  ->  ( w  = 
<. a ,  t >.  <->  w  =  <. <. r ,  s
>. ,  t >. ) )
171, 2, 4otth2 4004 . . . . . . . . . . . . . . . . . . 19  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  <->  ( x  =  r  /\  y  =  s  /\  z  =  t ) )
18 df-3an 922 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  r  /\  y  =  s  /\  z  =  t )  <->  ( ( x  =  r  /\  y  =  s )  /\  z  =  t ) )
1917, 18bitri 182 . . . . . . . . . . . . . . . . . 18  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  <->  ( (
x  =  r  /\  y  =  s )  /\  z  =  t
) )
2019anbi1i 446 . . . . . . . . . . . . . . . . 17  |-  ( (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph )  <->  ( (
( x  =  r  /\  y  =  s )  /\  z  =  t )  /\  ph ) )
21 anass 393 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( x  =  r  /\  y  =  s )  /\  z  =  t )  /\  ph )  <->  ( ( x  =  r  /\  y  =  s )  /\  ( z  =  t  /\  ph ) ) )
22 anass 393 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  =  r  /\  y  =  s )  /\  ( z  =  t  /\  ph ) )  <->  ( x  =  r  /\  (
y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
2320, 21, 223bitri 204 . . . . . . . . . . . . . . . 16  |-  ( (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph )  <->  ( x  =  r  /\  (
y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
24233exbii 1539 . . . . . . . . . . . . . . 15  |-  ( E. x E. y E. z ( <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>.  /\  ph )  <->  E. x E. y E. z ( x  =  r  /\  ( y  =  s  /\  ( z  =  t  /\  ph )
) ) )
25 oprabidlem 5567 . . . . . . . . . . . . . . . . . 18  |-  ( E. x E. z ( x  =  r  /\  ( y  =  s  /\  ( z  =  t  /\  ph )
) )  ->  E. x
( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
2625eximi 1532 . . . . . . . . . . . . . . . . 17  |-  ( E. y E. x E. z ( x  =  r  /\  ( y  =  s  /\  (
z  =  t  /\  ph ) ) )  ->  E. y E. x ( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
27 excom 1595 . . . . . . . . . . . . . . . . 17  |-  ( E. x E. y E. z ( x  =  r  /\  ( y  =  s  /\  (
z  =  t  /\  ph ) ) )  <->  E. y E. x E. z ( x  =  r  /\  ( y  =  s  /\  ( z  =  t  /\  ph )
) ) )
28 excom 1595 . . . . . . . . . . . . . . . . 17  |-  ( E. x E. y ( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) )  <->  E. y E. x ( x  =  r  /\  E. z
( y  =  s  /\  ( z  =  t  /\  ph )
) ) )
2926, 27, 283imtr4i 199 . . . . . . . . . . . . . . . 16  |-  ( E. x E. y E. z ( x  =  r  /\  ( y  =  s  /\  (
z  =  t  /\  ph ) ) )  ->  E. x E. y ( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
30 oprabidlem 5567 . . . . . . . . . . . . . . . 16  |-  ( E. x E. y ( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) )  ->  E. x ( x  =  r  /\  E. y E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
31 oprabidlem 5567 . . . . . . . . . . . . . . . . . 18  |-  ( E. y E. z ( y  =  s  /\  ( z  =  t  /\  ph ) )  ->  E. y ( y  =  s  /\  E. z ( z  =  t  /\  ph )
) )
3231anim2i 334 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  r  /\  E. y E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) )  ->  ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) ) )
3332eximi 1532 . . . . . . . . . . . . . . . 16  |-  ( E. x ( x  =  r  /\  E. y E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) )  ->  E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) ) )
3429, 30, 333syl 17 . . . . . . . . . . . . . . 15  |-  ( E. x E. y E. z ( x  =  r  /\  ( y  =  s  /\  (
z  =  t  /\  ph ) ) )  ->  E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) ) )
3524, 34sylbi 119 . . . . . . . . . . . . . 14  |-  ( E. x E. y E. z ( <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>.  /\  ph )  ->  E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) ) )
36 euequ1 2037 . . . . . . . . . . . . . . . . . . 19  |-  E! x  x  =  r
37 eupick 2021 . . . . . . . . . . . . . . . . . . 19  |-  ( ( E! x  x  =  r  /\  E. x
( x  =  r  /\  E. y ( y  =  s  /\  E. z ( z  =  t  /\  ph )
) ) )  -> 
( x  =  r  ->  E. y ( y  =  s  /\  E. z ( z  =  t  /\  ph )
) ) )
3836, 37mpan 415 . . . . . . . . . . . . . . . . . 18  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( x  =  r  ->  E. y ( y  =  s  /\  E. z ( z  =  t  /\  ph )
) ) )
39 euequ1 2037 . . . . . . . . . . . . . . . . . . . 20  |-  E! y  y  =  s
40 eupick 2021 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E! y  y  =  s  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( y  =  s  ->  E. z ( z  =  t  /\  ph ) ) )
4139, 40mpan 415 . . . . . . . . . . . . . . . . . . 19  |-  ( E. y ( y  =  s  /\  E. z
( z  =  t  /\  ph ) )  ->  ( y  =  s  ->  E. z
( z  =  t  /\  ph ) ) )
42 euequ1 2037 . . . . . . . . . . . . . . . . . . . 20  |-  E! z  z  =  t
43 eupick 2021 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E! z  z  =  t  /\  E. z
( z  =  t  /\  ph ) )  ->  ( z  =  t  ->  ph ) )
4442, 43mpan 415 . . . . . . . . . . . . . . . . . . 19  |-  ( E. z ( z  =  t  /\  ph )  ->  ( z  =  t  ->  ph ) )
4541, 44syl6 33 . . . . . . . . . . . . . . . . . 18  |-  ( E. y ( y  =  s  /\  E. z
( z  =  t  /\  ph ) )  ->  ( y  =  s  ->  ( z  =  t  ->  ph )
) )
4638, 45syl6 33 . . . . . . . . . . . . . . . . 17  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( x  =  r  ->  ( y  =  s  ->  ( z  =  t  ->  ph )
) ) )
47463impd 1153 . . . . . . . . . . . . . . . 16  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( ( x  =  r  /\  y  =  s  /\  z  =  t )  ->  ph )
)
4817, 47syl5bi 150 . . . . . . . . . . . . . . 15  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( <. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  ->  ph ) )
4948com12 30 . . . . . . . . . . . . . 14  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  ->  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  ->  ph ) )
5035, 49syl5 32 . . . . . . . . . . . . 13  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  ->  ( E. x E. y E. z ( <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>.  /\  ph )  ->  ph ) )
51 eqeq1 2088 . . . . . . . . . . . . . . 15  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( w  =  <. <.
x ,  y >. ,  z >.  <->  <. <. r ,  s >. ,  t
>.  =  <. <. x ,  y >. ,  z
>. ) )
52 eqcom 2084 . . . . . . . . . . . . . . 15  |-  ( <. <. r ,  s >. ,  t >.  =  <. <.
x ,  y >. ,  z >.  <->  <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>. )
5351, 52syl6bb 194 . . . . . . . . . . . . . 14  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( w  =  <. <.
x ,  y >. ,  z >.  <->  <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>. ) )
5453anbi1d 453 . . . . . . . . . . . . . . . 16  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ( <. <.
x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  /\  ph ) ) )
55543exbidv 1791 . . . . . . . . . . . . . . 15  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. y E. z (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph ) ) )
5655imbi1d 229 . . . . . . . . . . . . . 14  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( ( E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  ph )  <->  ( E. x E. y E. z (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph )  ->  ph )
) )
5753, 56imbi12d 232 . . . . . . . . . . . . 13  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
)  <->  ( <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>.  ->  ( E. x E. y E. z (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph )  ->  ph )
) ) )
5850, 57mpbiri 166 . . . . . . . . . . . 12  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) )
5916, 58syl6bi 161 . . . . . . . . . . 11  |-  ( a  =  <. r ,  s
>.  ->  ( w  = 
<. a ,  t >.  ->  ( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) ) )
6059adantr 270 . . . . . . . . . 10  |-  ( ( a  =  <. r ,  s >.  /\  <. r ,  s >.  =  <. x ,  y >. )  ->  ( w  =  <. a ,  t >.  ->  (
w  =  <. <. x ,  y >. ,  z
>.  ->  ( E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  ph ) ) ) )
6160exlimivv 1818 . . . . . . . . 9  |-  ( E. r E. s ( a  =  <. r ,  s >.  /\  <. r ,  s >.  =  <. x ,  y >. )  ->  ( w  =  <. a ,  t >.  ->  (
w  =  <. <. x ,  y >. ,  z
>.  ->  ( E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  ph ) ) ) )
6214, 61sylbi 119 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( w  = 
<. a ,  t >.  ->  ( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) ) )
6362com3l 80 . . . . . . 7  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
( a  =  <. x ,  y >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) ) )
6413, 63mpdd 40 . . . . . 6  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) )
6564adantr 270 . . . . 5  |-  ( ( w  =  <. a ,  t >.  /\  <. a ,  t >.  =  <. <.
x ,  y >. ,  z >. )  ->  ( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) )
6665exlimivv 1818 . . . 4  |-  ( E. a E. t ( w  =  <. a ,  t >.  /\  <. a ,  t >.  =  <. <.
x ,  y >. ,  z >. )  ->  ( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) )
678, 66mpcom 36 . . 3  |-  ( w  =  <. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
)
68 19.8a 1523 . . . . 5  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
69 19.8a 1523 . . . . 5  |-  ( E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
70 19.8a 1523 . . . . 5  |-  ( E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
7168, 69, 703syl 17 . . . 4  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
7271ex 113 . . 3  |-  ( w  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  ->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) ) )
7367, 72impbid 127 . 2  |-  ( w  =  <. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ph ) )
74 df-oprab 5547 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
756, 73, 74elab2 2742 1  |-  ( <. <. x ,  y >. ,  z >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434   E!weu 1942   _Vcvv 2602   <.cop 3409   {coprab 5544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-oprab 5547
This theorem is referenced by:  ssoprab2b  5593  ovid  5648  ovidig  5649  tposoprab  5929  xpcomco  6370
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