Theorem caovord2 7176

Description: Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.)

Hypotheses

Ref	Expression
caovord.1	⊢ 𝐴 ∈ V
caovord.2	⊢ 𝐵 ∈ V
caovord.3	⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
caovord2.3	⊢ 𝐶 ∈ V
caovord2.com	⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)

Assertion

Ref	Expression
caovord2	⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))

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

Proof of Theorem caovord2

Step	Hyp	Ref	Expression
1		caovord.1	. . 3 ⊢ 𝐴 ∈ V
2		caovord.2	. . 3 ⊢ 𝐵 ∈ V
3		caovord.3	. . 3 ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
4	1, 2, 3	caovord 7175	. 2 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
5		caovord2.3	. . . 4 ⊢ 𝐶 ∈ V
6		caovord2.com	. . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
7	5, 1, 6	caovcom 7161	. . 3 ⊢ (𝐶𝐹𝐴) = (𝐴𝐹𝐶)
8	5, 2, 6	caovcom 7161	. . 3 ⊢ (𝐶𝐹𝐵) = (𝐵𝐹𝐶)
9	7, 8	breq12i 4938	. 2 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))
10	4, 9	syl6bb 279	1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1507   ∈ wcel 2050  Vcvv 3415   class class class wbr 4929  (class class class)co 6976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-iota 6152  df-fv 6196  df-ov 6979

This theorem is referenced by:  caovord3  7177  genpnmax  10227  addclprlem1  10236  mulclprlem  10239  distrlem4pr  10246  ltexprlem6  10261  reclem3pr  10269  ltsosr  10314