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Theorem caovcld 6269
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 21 . 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 6268 . 2  |-  ( (
ph  /\  ( A  e.  C  /\  B  e.  D ) )  -> 
( A F B )  e.  E )
61, 2, 3, 5syl12anc 1183 1  |-  ( ph  ->  ( A F B )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727  (class class class)co 6110
This theorem is referenced by:  caovdir2d  6292  caov4d  6300  grprinvd  6315  climcn2  12417  plydivlem1  20241  plydivlem4  20244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-iota 5447  df-fv 5491  df-ov 6113
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