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Theorem caovdird 6998
Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovdirg.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
caovdird.2 (𝜑𝐴𝑆)
caovdird.3 (𝜑𝐵𝑆)
caovdird.4 (𝜑𝐶𝐾)
Assertion
Ref Expression
caovdird (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐻,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovdird
StepHypRef Expression
1 id 22 . 2 (𝜑𝜑)
2 caovdird.2 . 2 (𝜑𝐴𝑆)
3 caovdird.3 . 2 (𝜑𝐵𝑆)
4 caovdird.4 . 2 (𝜑𝐶𝐾)
5 caovdirg.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝐾)) → ((𝑥𝐹𝑦)𝐺𝑧) = ((𝑥𝐺𝑧)𝐻(𝑦𝐺𝑧)))
65caovdirg 6997 . 2 ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝐾)) → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
71, 2, 3, 4, 6syl13anc 1478 1 (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐻(𝐵𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  (class class class)co 6792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6795
This theorem is referenced by:  caofdir  7080
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