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Theorem caovordg 7611
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 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
Assertion
Ref Expression
caovordg ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (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
StepHypRef Expression
1 caovordg.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
21ralrimivvva 3204 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
3 breq1 5151 . . . 4 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
4 oveq2 7414 . . . . 5 (𝑥 = 𝐴 → (𝑧𝐹𝑥) = (𝑧𝐹𝐴))
54breq1d 5158 . . . 4 (𝑥 = 𝐴 → ((𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)))
63, 5bibi12d 346 . . 3 (𝑥 = 𝐴 → ((𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦))))
7 breq2 5152 . . . 4 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
8 oveq2 7414 . . . . 5 (𝑦 = 𝐵 → (𝑧𝐹𝑦) = (𝑧𝐹𝐵))
98breq2d 5160 . . . 4 (𝑦 = 𝐵 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))
107, 9bibi12d 346 . . 3 (𝑦 = 𝐵 → ((𝐴𝑅𝑦 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))))
11 oveq1 7413 . . . . 5 (𝑧 = 𝐶 → (𝑧𝐹𝐴) = (𝐶𝐹𝐴))
12 oveq1 7413 . . . . 5 (𝑧 = 𝐶 → (𝑧𝐹𝐵) = (𝐶𝐹𝐵))
1311, 12breq12d 5161 . . . 4 (𝑧 = 𝐶 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵) ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
1413bibi2d 343 . . 3 (𝑧 = 𝐶 → ((𝐴𝑅𝐵 ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)) ↔ (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
156, 10, 14rspc3v 3627 . 2 ((𝐴𝑆𝐵𝑆𝐶𝑆) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
162, 15mpan9 508 1 ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062   class class class wbr 5148  (class class class)co 7406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  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 6493  df-fv 6549  df-ov 7409
This theorem is referenced by:  caovordd  7612
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