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Theorem caovcld 7625
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 7624 . 2 ((𝜑 ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
61, 2, 3, 5syl12anc 837 1 (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  (class class class)co 7430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433
This theorem is referenced by:  caovdir2d  7648  caov4d  7656  climcn2  15625  grpinva  18699  plydivlem1  26349  plydivlem4  26352
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