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Theorem caovord 7632

Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.)

**Hypotheses**
Ref	Expression
caovord.1	⊢ 𝐴 ∈ V
caovord.2	⊢ 𝐵 ∈ V
caovord.3	⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))

**Assertion**
Ref	Expression
caovord	⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

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 caovord**
Step	Hyp	Ref	Expression
1		oveq1 7427	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝐹𝐴) = (𝐶𝐹𝐴))
2		oveq1 7427	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝐹𝐵) = (𝐶𝐹𝐵))
3	1, 2	breq12d 5115	. . 3 ⊢ (𝑧 = 𝐶 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵) ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
4	3	bibi2d 345	. 2 ⊢ (𝑧 = 𝐶 → ((𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)) ↔ (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
5		caovord.1	. . 3 ⊢ 𝐴 ∈ V
6		caovord.2	. . 3 ⊢ 𝐵 ∈ V
7		breq1 5105	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
8		oveq2 7428	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧𝐹𝑥) = (𝑧𝐹𝐴))
9	8	breq1d 5112	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)))
10	7, 9	bibi12d 348	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦))))
11		breq2 5106	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
12		oveq2 7428	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑧𝐹𝑦) = (𝑧𝐹𝐵))
13	12	breq2d 5114	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))
14	11, 13	bibi12d 348	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))))
15	10, 14	sylan9bb 519	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))))
16	15	imbi2d 343	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) ↔ (𝑧 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))))
17		caovord.3	. . 3 ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
18	5, 6, 16, 17	vtocl2 3526	. 2 ⊢ (𝑧 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))
19	4, 18	vtoclga 3536	1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 class class class wbr 5102 (class class class)co 7420

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 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-ext 2732

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 2099 df-clab 2739 df-cleq 2752 df-clel 2835 df-rab 3413 df-v 3452 df-ss 3896 df-dif 4047 df-un 4151 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-iota 6494 df-fv 6546 df-ov 7423

This theorem is used by: caovord2 7633 caovord3 7634 genpcl 11093

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