Theorem caovass 7607

Description: Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.)

Hypotheses

Ref	Expression
caovass.1	⊢ 𝐴 ∈ V
caovass.2	⊢ 𝐵 ∈ V
caovass.3	⊢ 𝐶 ∈ V
caovass.4	⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))

Assertion

Ref	Expression
caovass	⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem caovass

Step	Hyp	Ref	Expression
1		caovass.1	. 2 ⊢ 𝐴 ∈ V
2		caovass.2	. 2 ⊢ 𝐵 ∈ V
3		caovass.3	. 2 ⊢ 𝐶 ∈ V
4		tru 1544	. . 3 ⊢ ⊤
5		caovass.4	. . . . 5 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
6	5	a1i 11	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
7	6	caovassg 7605	. . 3 ⊢ ((⊤ ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
8	4, 7	mpan 690	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
9	1, 2, 3, 8	mp3an 1463	1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541   ∈ wcel 2108  Vcvv 3459  (class class class)co 7405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-ov 7408

This theorem is referenced by:  caov32  7634  caov12  7635  caov31  7636  caov13  7637  caov4  7638  caov411  7639  caovdilem  7642  caovmo  7644