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Theorem caovclg 6360
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 26-May-2014.)
Hypothesis
Ref Expression
caovclg.1  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
Assertion
Ref Expression
caovclg  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
Distinct variable groups:    x, y, A    y, B    x, C, y    x, D, y    x, E, y    ph, x, y   
x, F, y
Allowed substitution hint:    B( x)

Proof of Theorem caovclg
StepHypRef Expression
1 caovclg.1 . . 3  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
21ralrimivva 2908 . 2  |-  ( ph  ->  A. x  e.  C  A. y  e.  D  ( x F y )  e.  E )
3 oveq1 6202 . . . 4  |-  ( x  =  A  ->  (
x F y )  =  ( A F y ) )
43eleq1d 2521 . . 3  |-  ( x  =  A  ->  (
( x F y )  e.  E  <->  ( A F y )  e.  E ) )
5 oveq2 6203 . . . 4  |-  ( y  =  B  ->  ( A F y )  =  ( A F B ) )
65eleq1d 2521 . . 3  |-  ( y  =  B  ->  (
( A F y )  e.  E  <->  ( A F B )  e.  E
) )
74, 6rspc2v 3180 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ( x F y )  e.  E  ->  ( A F B )  e.  E ) )
82, 7mpan9 469 1  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2796  (class class class)co 6195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-iota 5484  df-fv 5529  df-ov 6198
This theorem is referenced by:  caovcld  6361  caovcl  6362  grprinvd  6407  seqcl2  11936  seqcaopr  11955  ercpbl  14601  imasmnd2  15572  gsumpropd2lem  15619  imasgrp2  15784  gsumzaddlem  16524  gsumzaddlemOLD  16526  imasrng  16829  divsrhm  17437  mplind  17703  plymullem  21812
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