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Theorem caovcld 6097
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovclg.1  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
caovcld.2  |-  ( ph  ->  A  e.  C )
caovcld.3  |-  ( ph  ->  B  e.  D )
Assertion
Ref Expression
caovcld  |-  ( ph  ->  ( 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 caovcld
StepHypRef Expression
1 id 19 . 2  |-  ( ph  ->  ph )
2 caovcld.2 . 2  |-  ( ph  ->  A  e.  C )
3 caovcld.3 . 2  |-  ( ph  ->  B  e.  D )
4 caovclg.1 . . 3  |-  ( (
ph  /\  ( x  e.  C  /\  y  e.  D ) )  -> 
( x F y )  e.  E )
54caovclg 6096 . 2  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
61, 2, 3, 5syl12anc 1180 1  |-  ( ph  ->  ( A F B )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710  (class class class)co 5942
This theorem is referenced by:  caovdir2d  6120  caov4d  6128  grprinvd  6143  climcn2  12156  plydivlem1  19771  plydivlem4  19774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-iota 5298  df-fv 5342  df-ov 5945
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