Theorem caovord2 7420

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 7419	. 2 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
5		caovord2.3	. . . 4 ⊢ 𝐶 ∈ V
6		caovord2.com	. . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
7	5, 1, 6	caovcom 7405	. . 3 ⊢ (𝐶𝐹𝐴) = (𝐴𝐹𝐶)
8	5, 2, 6	caovcom 7405	. . 3 ⊢ (𝐶𝐹𝐵) = (𝐵𝐹𝐶)
9	7, 8	breq12i 5062	. 2 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))
10	4, 9	bitrdi 290	1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1543   ∈ wcel 2110  Vcvv 3408   class class class wbr 5053  (class class class)co 7213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216

This theorem is referenced by:  caovord3  7421  genpnmax  10621  addclprlem1  10630  mulclprlem  10633  distrlem4pr  10640  ltexprlem6  10655  reclem3pr  10663  ltsosr  10708