Theorem caovordg 6200

Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996.) (Revised by Mario Carneiro, 30-Dec-2014.)

Hypothesis

Ref	Expression
caovordg.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))

Assertion

Ref	Expression
caovordg	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

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 caovordg

Step	Hyp	Ref	Expression
1		caovordg.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
2	1	ralrimivvva 2616	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
3		breq1 4096	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝐴𝑅𝑦))
4		oveq2 6036	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑧𝐹𝑥) = (𝑧𝐹𝐴))
5	4	breq1d 4103	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)))
6	3, 5	bibi12d 235	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦))))
7		breq2 4097	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝑅𝑦 ↔ 𝐴𝑅𝐵))
8		oveq2 6036	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑧𝐹𝑦) = (𝑧𝐹𝐵))
9	8	breq2d 4105	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))
10	7, 9	bibi12d 235	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))))
11		oveq1 6035	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧𝐹𝐴) = (𝐶𝐹𝐴))
12		oveq1 6035	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑧𝐹𝐵) = (𝐶𝐹𝐵))
13	11, 12	breq12d 4106	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵) ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
14	13	bibi2d 232	. . 3 ⊢ (𝑧 = 𝐶 → ((𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)) ↔ (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
15	6, 10, 14	rspc3v 2927	. 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
16	2, 15	mpan9 281	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511   class class class wbr 4093  (class class class)co 6028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  caovordd  6201