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Theorem caovdi 7369
 Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.)
Hypotheses
Ref Expression
caovdi.1 𝐴 ∈ V
caovdi.2 𝐵 ∈ V
caovdi.3 𝐶 ∈ V
caovdi.4 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))
Assertion
Ref Expression
caovdi (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧

Proof of Theorem caovdi
StepHypRef Expression
1 caovdi.1 . 2 𝐴 ∈ V
2 caovdi.2 . 2 𝐵 ∈ V
3 caovdi.3 . 2 𝐶 ∈ V
4 tru 1542 . . 3
5 caovdi.4 . . . . 5 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))
65a1i 11 . . . 4 ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
76caovdig 7364 . . 3 ((⊤ ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))
84, 7mpan 689 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))
91, 2, 3, 8mp3an 1458 1 (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  Vcvv 3409  (class class class)co 7156 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348  df-ov 7159 This theorem is referenced by:  caovdir  7384  caovlem2  7386
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