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Theorem oprabid 6097
Description: The law of concretion. Special case of Theorem 9.5 of [Quine] p. 61. (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 opex 4419 . 2  |-  <. <. x ,  y >. ,  z
>.  e.  _V
2 opex 4419 . . . . . 6  |-  <. x ,  y >.  e.  _V
3 vex 2951 . . . . . 6  |-  z  e. 
_V
42, 3eqvinop 4433 . . . . 5  |-  ( w  =  <. <. x ,  y
>. ,  z >.  <->  E. a E. t ( w  =  <. a ,  t
>.  /\  <. a ,  t
>.  =  <. <. x ,  y >. ,  z
>. ) )
54biimpi 187 . . . 4  |-  ( w  =  <. <. x ,  y
>. ,  z >.  ->  E. a E. t ( w  =  <. a ,  t >.  /\  <. a ,  t >.  =  <. <.
x ,  y >. ,  z >. )
)
6 eqeq1 2441 . . . . . . . 8  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  <->  <. a ,  t >.  =  <. <.
x ,  y >. ,  z >. )
)
7 vex 2951 . . . . . . . . 9  |-  a  e. 
_V
8 vex 2951 . . . . . . . . 9  |-  t  e. 
_V
97, 8opth1 4426 . . . . . . . 8  |-  ( <.
a ,  t >.  =  <. <. x ,  y
>. ,  z >.  -> 
a  =  <. x ,  y >. )
106, 9syl6bi 220 . . . . . . 7  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
a  =  <. x ,  y >. )
)
11 vex 2951 . . . . . . . . . 10  |-  x  e. 
_V
12 vex 2951 . . . . . . . . . 10  |-  y  e. 
_V
1311, 12eqvinop 4433 . . . . . . . . 9  |-  ( a  =  <. x ,  y
>. 
<->  E. r E. s
( a  =  <. r ,  s >.  /\  <. r ,  s >.  =  <. x ,  y >. )
)
14 opeq1 3976 . . . . . . . . . . . . 13  |-  ( a  =  <. r ,  s
>.  ->  <. a ,  t
>.  =  <. <. r ,  s >. ,  t
>. )
1514eqeq2d 2446 . . . . . . . . . . . 12  |-  ( a  =  <. r ,  s
>.  ->  ( w  = 
<. a ,  t >.  <->  w  =  <. <. r ,  s
>. ,  t >. ) )
1611, 12, 3otth2 4431 . . . . . . . . . . . . . . . . . . 19  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  <->  ( x  =  r  /\  y  =  s  /\  z  =  t ) )
17 df-3an 938 . . . . . . . . . . . . . . . . . . 19  |-  ( ( x  =  r  /\  y  =  s  /\  z  =  t )  <->  ( ( x  =  r  /\  y  =  s )  /\  z  =  t ) )
1816, 17bitri 241 . . . . . . . . . . . . . . . . . 18  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  <->  ( (
x  =  r  /\  y  =  s )  /\  z  =  t
) )
1918anbi1i 677 . . . . . . . . . . . . . . . . 17  |-  ( (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph )  <->  ( (
( x  =  r  /\  y  =  s )  /\  z  =  t )  /\  ph ) )
20 anass 631 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( x  =  r  /\  y  =  s )  /\  z  =  t )  /\  ph )  <->  ( ( x  =  r  /\  y  =  s )  /\  ( z  =  t  /\  ph ) ) )
21 anass 631 . . . . . . . . . . . . . . . . 17  |-  ( ( ( x  =  r  /\  y  =  s )  /\  ( z  =  t  /\  ph ) )  <->  ( x  =  r  /\  (
y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
2219, 20, 213bitri 263 . . . . . . . . . . . . . . . 16  |-  ( (
<. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  /\ 
ph )  <->  ( x  =  r  /\  (
y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
23223exbii 1594 . . . . . . . . . . . . . . 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 )
) ) )
24 nfcvf2 2594 . . . . . . . . . . . . . . . . . . . 20  |-  ( -. 
A. x  x  =  z  ->  F/_ z x )
25 nfcvd 2572 . . . . . . . . . . . . . . . . . . . 20  |-  ( -. 
A. x  x  =  z  ->  F/_ z r )
2624, 25nfeqd 2585 . . . . . . . . . . . . . . . . . . 19  |-  ( -. 
A. x  x  =  z  ->  F/ z  x  =  r )
2726exdistrf 2062 . . . . . . . . . . . . . . . . . 18  |-  ( E. x E. z ( x  =  r  /\  ( y  =  s  /\  ( z  =  t  /\  ph )
) )  ->  E. x
( x  =  r  /\  E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) ) )
2827eximi 1585 . . . . . . . . . . . . . . . . 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 ) ) ) )
29 excom 1756 . . . . . . . . . . . . . . . . 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 )
) ) )
30 excom 1756 . . . . . . . . . . . . . . . . 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 )
) ) )
3128, 29, 303imtr4i 258 . . . . . . . . . . . . . . . 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 ) ) ) )
32 nfcvf2 2594 . . . . . . . . . . . . . . . . . 18  |-  ( -. 
A. x  x  =  y  ->  F/_ y x )
33 nfcvd 2572 . . . . . . . . . . . . . . . . . 18  |-  ( -. 
A. x  x  =  y  ->  F/_ y r )
3432, 33nfeqd 2585 . . . . . . . . . . . . . . . . 17  |-  ( -. 
A. x  x  =  y  ->  F/ y  x  =  r )
3534exdistrf 2062 . . . . . . . . . . . . . . . 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 ) ) ) )
36 nfcvf2 2594 . . . . . . . . . . . . . . . . . . . 20  |-  ( -. 
A. y  y  =  z  ->  F/_ z y )
37 nfcvd 2572 . . . . . . . . . . . . . . . . . . . 20  |-  ( -. 
A. y  y  =  z  ->  F/_ z s )
3836, 37nfeqd 2585 . . . . . . . . . . . . . . . . . . 19  |-  ( -. 
A. y  y  =  z  ->  F/ z 
y  =  s )
3938exdistrf 2062 . . . . . . . . . . . . . . . . . 18  |-  ( E. y E. z ( y  =  s  /\  ( z  =  t  /\  ph ) )  ->  E. y ( y  =  s  /\  E. z ( z  =  t  /\  ph )
) )
4039anim2i 553 . . . . . . . . . . . . . . . . 17  |-  ( ( x  =  r  /\  E. y E. z ( y  =  s  /\  ( z  =  t  /\  ph ) ) )  ->  ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) ) )
4140eximi 1585 . . . . . . . . . . . . . . . 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 ) ) ) )
4231, 35, 413syl 19 . . . . . . . . . . . . . . 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 ) ) ) )
4323, 42sylbi 188 . . . . . . . . . . . . . 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 ) ) ) )
44 euequ1 2368 . . . . . . . . . . . . . . . . . . 19  |-  E! x  x  =  r
45 eupick 2343 . . . . . . . . . . . . . . . . . . 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 )
) ) )
4644, 45mpan 652 . . . . . . . . . . . . . . . . . 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 )
) ) )
47 euequ1 2368 . . . . . . . . . . . . . . . . . . . 20  |-  E! y  y  =  s
48 eupick 2343 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E! y  y  =  s  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( y  =  s  ->  E. z ( z  =  t  /\  ph ) ) )
4947, 48mpan 652 . . . . . . . . . . . . . . . . . . 19  |-  ( E. y ( y  =  s  /\  E. z
( z  =  t  /\  ph ) )  ->  ( y  =  s  ->  E. z
( z  =  t  /\  ph ) ) )
50 euequ1 2368 . . . . . . . . . . . . . . . . . . . 20  |-  E! z  z  =  t
51 eupick 2343 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( E! z  z  =  t  /\  E. z
( z  =  t  /\  ph ) )  ->  ( z  =  t  ->  ph ) )
5250, 51mpan 652 . . . . . . . . . . . . . . . . . . 19  |-  ( E. z ( z  =  t  /\  ph )  ->  ( z  =  t  ->  ph ) )
5349, 52syl6 31 . . . . . . . . . . . . . . . . . 18  |-  ( E. y ( y  =  s  /\  E. z
( z  =  t  /\  ph ) )  ->  ( y  =  s  ->  ( z  =  t  ->  ph )
) )
5446, 53syl6 31 . . . . . . . . . . . . . . . . 17  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( x  =  r  ->  ( y  =  s  ->  ( z  =  t  ->  ph )
) ) )
55543impd 1167 . . . . . . . . . . . . . . . 16  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( ( x  =  r  /\  y  =  s  /\  z  =  t )  ->  ph )
)
5616, 55syl5bi 209 . . . . . . . . . . . . . . 15  |-  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  -> 
( <. <. x ,  y
>. ,  z >.  = 
<. <. r ,  s
>. ,  t >.  ->  ph ) )
5756com12 29 . . . . . . . . . . . . . 14  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  ->  ( E. x ( x  =  r  /\  E. y
( y  =  s  /\  E. z ( z  =  t  /\  ph ) ) )  ->  ph ) )
5843, 57syl5 30 . . . . . . . . . . . . 13  |-  ( <. <. x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  ->  ( E. x E. y E. z ( <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>.  /\  ph )  ->  ph ) )
59 eqeq1 2441 . . . . . . . . . . . . . . 15  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( w  =  <. <.
x ,  y >. ,  z >.  <->  <. <. r ,  s >. ,  t
>.  =  <. <. x ,  y >. ,  z
>. ) )
60 eqcom 2437 . . . . . . . . . . . . . . 15  |-  ( <. <. r ,  s >. ,  t >.  =  <. <.
x ,  y >. ,  z >.  <->  <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>. )
6159, 60syl6bb 253 . . . . . . . . . . . . . 14  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( w  =  <. <.
x ,  y >. ,  z >.  <->  <. <. x ,  y >. ,  z
>.  =  <. <. r ,  s >. ,  t
>. ) )
6261anbi1d 686 . . . . . . . . . . . . . . . 16  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ( <. <.
x ,  y >. ,  z >.  =  <. <.
r ,  s >. ,  t >.  /\  ph ) ) )
63623exbidv 1639 . . . . . . . . . . . . . . 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 ) ) )
6463imbi1d 309 . . . . . . . . . . . . . 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 )
) )
6561, 64imbi12d 312 . . . . . . . . . . . . 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 )
) ) )
6658, 65mpbiri 225 . . . . . . . . . . . 12  |-  ( w  =  <. <. r ,  s
>. ,  t >.  -> 
( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) )
6715, 66syl6bi 220 . . . . . . . . . . 11  |-  ( a  =  <. r ,  s
>.  ->  ( w  = 
<. a ,  t >.  ->  ( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) ) )
6867adantr 452 . . . . . . . . . 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 ) ) ) )
6968exlimivv 1645 . . . . . . . . 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 ) ) ) )
7013, 69sylbi 188 . . . . . . . 8  |-  ( a  =  <. x ,  y
>.  ->  ( w  = 
<. a ,  t >.  ->  ( w  =  <. <.
x ,  y >. ,  z >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) ) )
7170com3l 77 . . . . . . 7  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
( a  =  <. x ,  y >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) ) )
7210, 71mpdd 38 . . . . . 6  |-  ( w  =  <. a ,  t
>.  ->  ( w  = 
<. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
) )
7372adantr 452 . . . . 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 )
) )
7473exlimivv 1645 . . . 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 )
) )
755, 74mpcom 34 . . 3  |-  ( w  =  <. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  ph )
)
76 19.8a 1762 . . . . 5  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
77 19.8a 1762 . . . . 5  |-  ( E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  ->  E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
78 19.8a 1762 . . . . 5  |-  ( E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
7976, 77, 783syl 19 . . . 4  |-  ( ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph )  ->  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) )
8079ex 424 . . 3  |-  ( w  =  <. <. x ,  y
>. ,  z >.  -> 
( ph  ->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) ) )
8175, 80impbid 184 . 2  |-  ( w  =  <. <. x ,  y
>. ,  z >.  -> 
( E. x E. y E. z ( w  =  <. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  ph ) )
82 df-oprab 6077 . 2  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
831, 81, 82elab2 3077 1  |-  ( <. <. x ,  y >. ,  z >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   E!weu 2280   <.cop 3809   {coprab 6074
This theorem is referenced by:  ssoprab2b  6123  ovid  6182  ovidig  6183  tposoprab  6507  xpcomco  7190
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-oprab 6077
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