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Theorem eloprabg 6363
Description: The law of concretion for operation class abstraction. Compare elopab 4744. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
eloprabg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
eloprabg.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
eloprabg.3  |-  ( z  =  C  ->  ( ch 
<->  th ) )
Assertion
Ref Expression
eloprabg  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  th )
)
Distinct variable groups:    x, y,
z, A    x, B, y, z    x, C, y, z    th, x, y, z
Allowed substitution hints:    ph( x, y, z)    ps( x, y, z)    ch( x, y, z)    V( x, y, z)    W( x, y, z)    X( x, y, z)

Proof of Theorem eloprabg
StepHypRef Expression
1 eloprabg.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
2 eloprabg.2 . . 3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
3 eloprabg.3 . . 3  |-  ( z  =  C  ->  ( ch 
<->  th ) )
41, 2, 3syl3an9b 1295 . 2  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  th )
)
54eloprabga 6362 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  th )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1398    e. wcel 1823   <.cop 4022   {coprab 6271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-oprab 6274
This theorem is referenced by:  ov  6395  ovg  6414  brbtwn  24407  isnvlem  25704  isphg  25933  fvtransport  29913  brcolinear2  29939  colineardim1  29942  fvray  30022  fvline  30025
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