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Theorem caovordig 5943
Description: Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
Hypothesis
Ref Expression
caovordig.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
Assertion
Ref Expression
caovordig ((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 caovordig
StepHypRef Expression
1 caovordig.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
21ralrimivvva 2518 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆𝑧𝑆 (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
3 breq1 3939 . . . 4 (𝑥 = 𝐴 → (𝑥𝑅𝑦𝐴𝑅𝑦))
4 oveq2 5789 . . . . 5 (𝑥 = 𝐴 → (𝑧𝐹𝑥) = (𝑧𝐹𝐴))
54breq1d 3946 . . . 4 (𝑥 = 𝐴 → ((𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)))
63, 5imbi12d 233 . . 3 (𝑥 = 𝐴 → ((𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝑦 → (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦))))
7 breq2 3940 . . . 4 (𝑦 = 𝐵 → (𝐴𝑅𝑦𝐴𝑅𝐵))
8 oveq2 5789 . . . . 5 (𝑦 = 𝐵 → (𝑧𝐹𝑦) = (𝑧𝐹𝐵))
98breq2d 3948 . . . 4 (𝑦 = 𝐵 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦) ↔ (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)))
107, 9imbi12d 233 . . 3 (𝑦 = 𝐵 → ((𝐴𝑅𝑦 → (𝑧𝐹𝐴)𝑅(𝑧𝐹𝑦)) ↔ (𝐴𝑅𝐵 → (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵))))
11 oveq1 5788 . . . . 5 (𝑧 = 𝐶 → (𝑧𝐹𝐴) = (𝐶𝐹𝐴))
12 oveq1 5788 . . . . 5 (𝑧 = 𝐶 → (𝑧𝐹𝐵) = (𝐶𝐹𝐵))
1311, 12breq12d 3949 . . . 4 (𝑧 = 𝐶 → ((𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵) ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
1413imbi2d 229 . . 3 (𝑧 = 𝐶 → ((𝐴𝑅𝐵 → (𝑧𝐹𝐴)𝑅(𝑧𝐹𝐵)) ↔ (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
156, 10, 14rspc3v 2808 . 2 ((𝐴𝑆𝐵𝑆𝐶𝑆) → (∀𝑥𝑆𝑦𝑆𝑧𝑆 (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)) → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))))
162, 15mpan9 279 1 ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1332  wcel 1481  wral 2417   class class class wbr 3936  (class class class)co 5781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-iota 5095  df-fv 5138  df-ov 5784
This theorem is referenced by:  caovordid  5944
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