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Theorem eloprabga 5934
Description: The law of concretion for operation class abstraction. Compare elopab 4272. (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 2796 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 elex 2796 . 2  |-  ( B  e.  W  ->  B  e.  _V )
3 elex 2796 . 2  |-  ( C  e.  X  ->  C  e.  _V )
4 opex 4237 . . 3  |-  <. <. A ,  B >. ,  C >.  e. 
_V
5 simpr 447 . . . . . . . . . 10  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  w  =  <. <. A ,  B >. ,  C >. )
65eqeq1d 2291 . . . . . . . . 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 2285 . . . . . . . . . 10  |-  ( <. <. A ,  B >. ,  C >.  =  <. <.
x ,  y >. ,  z >.  <->  <. <. x ,  y >. ,  z
>.  =  <. <. A ,  B >. ,  C >. )
8 vex 2791 . . . . . . . . . . 11  |-  x  e. 
_V
9 vex 2791 . . . . . . . . . . 11  |-  y  e. 
_V
10 vex 2791 . . . . . . . . . . 11  |-  z  e. 
_V
118, 9, 10otth2 4249 . . . . . . . . . 10  |-  ( <. <. x ,  y >. ,  z >.  =  <. <. A ,  B >. ,  C >.  <->  ( x  =  A  /\  y  =  B  /\  z  =  C ) )
127, 11bitri 240 . . . . . . . . 9  |-  ( <. <. A ,  B >. ,  C >.  =  <. <.
x ,  y >. ,  z >.  <->  ( x  =  A  /\  y  =  B  /\  z  =  C ) )
136, 12syl6bb 252 . . . . . . . 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 685 . . . . . . 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 618 . . . . . . 7  |-  ( ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ph )  <->  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  /\  ps )
)
1714, 16syl6bb 252 . . . . . 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 1615 . . . . 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 5862 . . . . . . . . 9  |-  { <. <.
x ,  y >. ,  z >.  |  ph }  =  { w  |  E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph ) }
2019eleq2i 2347 . . . . . . . 8  |-  ( w  e.  { <. <. x ,  y >. ,  z
>.  |  ph }  <->  w  e.  { w  |  E. x E. y E. z ( w  =  <. <. x ,  y >. ,  z
>.  /\  ph ) } )
21 abid 2271 . . . . . . . 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 241 . . . . . . 7  |-  ( E. x E. y E. z ( w  = 
<. <. x ,  y
>. ,  z >.  /\ 
ph )  <->  w  e.  {
<. <. x ,  y
>. ,  z >.  | 
ph } )
23 eleq1 2343 . . . . . . 7  |-  ( w  =  <. <. A ,  B >. ,  C >.  ->  (
w  e.  { <. <.
x ,  y >. ,  z >.  |  ph } 
<-> 
<. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph } ) )
2422, 23syl5bb 248 . . . . . 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 452 . . . . 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 2798 . . . . . . . . . 10  |-  ( A  e.  _V  ->  E. x  x  =  A )
27 elisset 2798 . . . . . . . . . 10  |-  ( B  e.  _V  ->  E. y 
y  =  B )
28 elisset 2798 . . . . . . . . . 10  |-  ( C  e.  _V  ->  E. z 
z  =  C )
2926, 27, 283anim123i 1137 . . . . . . . . 9  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( E. x  x  =  A  /\  E. y  y  =  B  /\  E. z  z  =  C
) )
30 eeeanv 1855 . . . . . . . . 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 203 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  E. x E. y E. z ( x  =  A  /\  y  =  B  /\  z  =  C )
)
3231biantrurd 494 . . . . . . 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 1844 . . . . . . 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 255 . . . . . 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 451 . . . . 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 274 . . . 4  |-  ( ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  /\  w  =  <. <. A ,  B >. ,  C >. )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
3736expcom 424 . . 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 2857 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  C  e.  _V )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  ps )
)
391, 2, 3, 38syl3an 1224 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 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788   <.cop 3643   {coprab 5859
This theorem is referenced by:  eloprabg  5935  ovigg  5968  vdwpc  13027
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-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  df-oprab 5862
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