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Theorem eloprabg 5935
Description: The law of concretion for operation class abstraction. Compare elopab 4272. (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 1250 . 2  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  th )
)
54eloprabga 5934 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  C  e.  X )  ->  ( <. <. A ,  B >. ,  C >.  e.  { <. <. x ,  y
>. ,  z >.  | 
ph }  <->  th )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934    = wceq 1623    e. wcel 1684   <.cop 3643   {coprab 5859
This theorem is referenced by:  ov  5967  ovg  5986  isnvlem  21166  isphg  21395  brbtwn  24527  fvtransport  24655  brcolinear2  24681  colineardim1  24684  fvray  24764  fvline  24767  ismgra  25710  prismorcset  25914
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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