MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eloprabg Structured version   Unicode version

Theorem eloprabg 6177
Description: The law of concretion for operation class abstraction. Compare elopab 4596. (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 1287 . 2  |-  ( ( x  =  A  /\  y  =  B  /\  z  =  C )  ->  ( ph  <->  th )
)
54eloprabga 6176 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 965    = wceq 1369    e. wcel 1756   <.cop 3882   {coprab 6091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-oprab 6094
This theorem is referenced by:  ov  6209  ovg  6228  brbtwn  23144  isnvlem  23987  isphg  24216  fvtransport  28062  brcolinear2  28088  colineardim1  28091  fvray  28171  fvline  28174
  Copyright terms: Public domain W3C validator