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Theorem caovcld 7551
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 7550 . 2 ((𝜑 ∧ (𝐴𝐶𝐵𝐷)) → (𝐴𝐹𝐵) ∈ 𝐸)
61, 2, 3, 5syl12anc 836 1 (𝜑 → (𝐴𝐹𝐵) ∈ 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  (class class class)co 7361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364
This theorem is referenced by:  caovdir2d  7574  caov4d  7582  climcn2  15484  grprinvd  18537  plydivlem1  25676  plydivlem4  25679
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