Theorem caovord2d 7615

Description: Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

Ref	Expression
caovordg.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
caovordd.2	⊢ (𝜑 → 𝐴 ∈ 𝑆)
caovordd.3	⊢ (𝜑 → 𝐵 ∈ 𝑆)
caovordd.4	⊢ (𝜑 → 𝐶 ∈ 𝑆)
caovord2d.com	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))

Assertion

Ref	Expression
caovord2d	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))

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

Proof of Theorem caovord2d

Step	Hyp	Ref	Expression
1		caovordg.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
2		caovordd.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
3		caovordd.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆)
4		caovordd.4	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑆)
5	1, 2, 3, 4	caovordd 7614	. 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
6		caovord2d.com	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
7	6, 4, 2	caovcomd 7602	. . 3 ⊢ (𝜑 → (𝐶𝐹𝐴) = (𝐴𝐹𝐶))
8	6, 4, 3	caovcomd 7602	. . 3 ⊢ (𝜑 → (𝐶𝐹𝐵) = (𝐵𝐹𝐶))
9	7, 8	breq12d 5161	. 2 ⊢ (𝜑 → ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))
10	5, 9	bitrd 278	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5148  (class class class)co 7408

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7411

This theorem is referenced by:  caovord3d  7616