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Theorem caovord2 7351
 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 (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
caovord2.3 𝐶 ∈ V
caovord2.com (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
Assertion
Ref Expression
caovord2 (𝐶𝑆 → (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
StepHypRef Expression
1 caovord.1 . . 3 𝐴 ∈ V
2 caovord.2 . . 3 𝐵 ∈ V
3 caovord.3 . . 3 (𝑧𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
41, 2, 3caovord 7350 . 2 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
5 caovord2.3 . . . 4 𝐶 ∈ V
6 caovord2.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
75, 1, 6caovcom 7336 . . 3 (𝐶𝐹𝐴) = (𝐴𝐹𝐶)
85, 2, 6caovcom 7336 . . 3 (𝐶𝐹𝐵) = (𝐵𝐹𝐶)
97, 8breq12i 5043 . 2 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))
104, 9syl6bb 290 1 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  Vcvv 3442   class class class wbr 5034  (class class class)co 7145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-iota 6291  df-fv 6340  df-ov 7148 This theorem is referenced by:  caovord3  7352  genpnmax  10436  addclprlem1  10445  mulclprlem  10448  distrlem4pr  10455  ltexprlem6  10470  reclem3pr  10478  ltsosr  10523
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