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Theorem eloprabga 6100
Description: The law of concretion for operation class abstraction. Compare elopab 4404. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
eloprabga.1  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
eloprabga  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    ps, x, y, z
Allowed substitution hints:    ph( x, y, z)    V( x, y, z)    W( x, y, z)    X( x, y, z)

Proof of Theorem eloprabga
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 2908 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2908 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 elex 2908 . 2  |-  ( C  e.  X  ->  C  e.  _V )
4 opex 4369 . . 3  |-  <. <. A ,  B >. ,  C >.  e. 
_V
5 simpr 448 . . . . . . . . . 10  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  w  =  <. <. A ,  B >. ,  C >. )
65eqeq1d 2396 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  (
w  =  <. <. x ,  y >. ,  z
>. 
<-> 
<. <. A ,  B >. ,  C >.  =  <. <.
x ,  y >. ,  z >. )
)
7 eqcom 2390 . . . . . . . . . 10  |-  ( <. <. A ,  B >. ,  C >.  =  <. <.
x ,  y >. ,  z >.  <->  <. <. x ,  y >. ,  z
>.  =  <. <. A ,  B >. ,  C >. )
8 vex 2903 . . . . . . . . . . 11  |-  x  e. 
_V
9 vex 2903 . . . . . . . . . . 11  |-  y  e. 
_V
10 vex 2903 . . . . . . . . . . 11  |-  z  e. 
_V
118, 9, 10otth2 4381 . . . . . . . . . 10  |-  ( <. <. x ,  y >. ,  z >.  =  <. <. A ,  B >. ,  C >.  <->  ( x  =  A  /\  y  =  B  /\  z  =  C ) )
127, 11bitri 241 . . . . . . . . 9  |-  ( <. <. A ,  B >. ,  C >.  =  <. <.
x ,  y >. ,  z >.  <->  ( x  =  A  /\  y  =  B  /\  z  =  C ) )
136, 12syl6bb 253 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  (
w  =  <. <. x ,  y >. ,  z
>. 
<->  ( x  =  A  /\  y  =  B  /\  z  =  C ) ) )
1413anbi1d 686 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  (
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph ) ) )
15 eloprabga.1 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  ps )
)
1615pm5.32i 619 . . . . . . 7  |-  ( ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph )  <->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps )
)
1714, 16syl6bb 253 . . . . . 6  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  (
( w  =  <. <.
x ,  y >. ,  z >.  /\  ph ) 
<->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) ) )
18173exbidv 1636 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps )
) )
19 df-oprab 6025 . . . . . . . . 9  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
2019eleq2i 2452 . . . . . . . 8  |-  ( w  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  <->  w  e.  { w  |  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) } )
21 abid 2376 . . . . . . . 8  |-  ( w  e.  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }  <->  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) )
2220, 21bitr2i 242 . . . . . . 7  |-  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  w  e.  {
<. <. x ,  y
>. ,  z >.  | 
ph } )
23 eleq1 2448 . . . . . . 7  |-  ( w  =  <. <. A ,  B >. ,  C >.  ->  (
w  e.  { <. <.
x ,  y >. ,  z >.  |  ph } 
<-> 
<. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph } ) )
2422, 23syl5bb 249 . . . . . 6  |-  ( w  =  <. <. A ,  B >. ,  C >.  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  <. <. A ,  B >. ,  C >.  e. 
{ <. <. x ,  y
>. ,  z >.  | 
ph } ) )
2524adantl 453 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  <. <. A ,  B >. ,  C >.  e. 
{ <. <. x ,  y
>. ,  z >.  | 
ph } ) )
26 elisset 2910 . . . . . . . . . 10  |-  ( A  e.  _V  ->  E. x  x  =  A )
27 elisset 2910 . . . . . . . . . 10  |-  ( B  e.  _V  ->  E. y 
y  =  B )
28 elisset 2910 . . . . . . . . . 10  |-  ( C  e.  _V  ->  E. z 
z  =  C )
2926, 27, 283anim123i 1139 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( E. x  x  =  A  /\  E. y  y  =  B  /\  E. z  z  =  C
) )
30 eeeanv 1927 . . . . . . . . 9  |-  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  <->  ( E. x  x  =  A  /\  E. y  y  =  B  /\  E. z 
z  =  C ) )
3129, 30sylibr 204 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )
)
3231biantrurd 495 . . . . . . 7  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( ps 
<->  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) ) )
33 19.41vvv 1915 . . . . . . 7  |-  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) 
<->  ( E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) )
3432, 33syl6rbbr 256 . . . . . 6  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) 
<->  ps ) )
3534adantr 452 . . . . 5  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  ( E. x E. y E. z ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps ) 
<->  ps ) )
3618, 25, 353bitr3d 275 . . . 4  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
3736expcom 425 . . 3  |-  ( w  =  <. <. A ,  B >. ,  C >.  ->  (
( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
) )
384, 37vtocle 2969 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
391, 2, 3, 38syl3an 1226 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374   _Vcvv 2900   <.cop 3761   {coprab 6022
This theorem is referenced by:  eloprabg  6101  ovigg  6134  vdwpc  13276
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-oprab 6025
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