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Theorem caovcl 5623
 Description: Convert an operation closure law to class notation. (Contributed by set.mm contributors, 4-Aug-1995.) (Revised by set.mm contributors, 26-May-2014.)
Hypothesis
Ref Expression
caovcl.1 ((x S y S) → (xFy) S)
Assertion
Ref Expression
caovcl ((A S B S) → (AFB) 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 1321 . 2
2 caovcl.1 . . . 4 ((x S y S) → (xFy) S)
32adantl 452 . . 3 (( ⊤ (x S y S)) → (xFy) S)
43caovcld 5622 . 2 (( ⊤ (A S B S)) → (AFB) S)
51, 4mpan 651 1 ((A S B S) → (AFB) S)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ⊤ wtru 1316   ∈ wcel 1710  (class class class)co 5525 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 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795  df-ov 5526 This theorem is referenced by: (None)
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