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Theorem eloprabg 5623
Description: The law of concretion for operation class abstraction. Compare elopab 4021. (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 1242 . 2  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  th )
)
54eloprabga 5622 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 103    /\ w3a 920    = wceq 1285    e. wcel 1434   <.cop 3409   {coprab 5544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-oprab 5547
This theorem is referenced by:  ov  5651  ovg  5670
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