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Theorem caovcl 6014
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 1312 . 2  |-  T.
2 caovcl.1 . . . 4  |-  ( ( x  e.  S  /\  y  e.  S )  ->  ( x F y )  e.  S )
32adantl 452 . . 3  |-  ( (  T.  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x F y )  e.  S )
43caovclg 6012 . 2  |-  ( (  T.  /\  ( A  e.  S  /\  B  e.  S ) )  -> 
( A F B )  e.  S )
51, 4mpan 651 1  |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    T. wtru 1307    e. wcel 1684  (class class class)co 5858
This theorem is referenced by:  ecopovtrn  6761  eceqoveq  6763  genpss  8628  genpnnp  8629  genpass  8633  expcllem  11114  txlly  17330  txnlly  17331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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