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Theorem caovcl 6464
Description: Convert an operation closure law to class notation. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)
Hypothesis
Ref Expression
caovcl.1  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  e.  S )
Assertion
Ref Expression
caovcl  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )
Distinct variable groups:    x, y, A    y, B    x, F, y    x, S, y
Allowed substitution hint:    B( x)

Proof of Theorem caovcl
StepHypRef Expression
1 tru 1383 . 2  |- T.
2 caovcl.1 . . . 4  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  e.  S )
32adantl 466 . . 3  |-  ( ( T.  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
43caovclg 6462 . 2  |-  ( ( T.  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  e.  S )
51, 4mpan 670 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   T. wtru 1380    e. wcel 1767  (class class class)co 6295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298
This theorem is referenced by:  ecopovtrn  7426  eceqoveq  7428  genpss  9394  genpnnp  9395  genpass  9399  expcllem  12157  txlly  20005  txnlly  20006
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