Theorem caovdi 7647

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

Step	Hyp	Ref	Expression
1		caovdi.1	. 2 ⊢ 𝐴 ∈ V
2		caovdi.2	. 2 ⊢ 𝐵 ∈ V
3		caovdi.3	. 2 ⊢ 𝐶 ∈ V
4		tru 1573	. . 3 ⊢ ⊤
5		caovdi.4	. . . . 5 ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))
6	5	a1i 11	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)))
7	6	caovdig 7642	. . 3 ⊢ ((⊤ ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))
8	4, 7	mpan 703	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)))
9	1, 2, 3, 8	mp3an 1490	1 ⊢ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ⊤wtru 1570   ∈ wcel 2146  Vcvv 3453  (class class class)co 7427

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-iota 6500  df-fv 6554  df-ov 7430

This theorem is used by:  caovdir  7662  caovlem2  7664