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Theorem caovcld 7544
Description: Convert an operation closure law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovclg.1 ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)
caovcld.2 (𝜑𝐴𝐶)
caovcld.3 (𝜑𝐵𝐷)
Assertion
Ref Expression
caovcld (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐸,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem caovcld
StepHypRef Expression
1 id 22 . 2 (𝜑𝜑)
2 caovcld.2 . 2 (𝜑𝐴𝐶)
3 caovcld.3 . 2 (𝜑𝐵𝐷)
4 caovclg.1 . . 3 ((𝜑 ∧ (𝑥𝐶𝑦𝐷)) → (𝑥𝐹𝑦) ∈ 𝐸)
54caovclg 7543 . 2 ((𝜑 ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
61, 2, 3, 5syl12anc 835 1 (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  (class class class)co 7354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3064  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-iota 6446  df-fv 6502  df-ov 7357
This theorem is referenced by:  caovdir2d  7567  caov4d  7575  climcn2  15472  grprinvd  18526  plydivlem1  25649  plydivlem4  25652
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