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Theorem caovord2 6811
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 6810 . 2 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
5 caovord2.3 . . . 4 𝐶 ∈ V
6 caovord2.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
75, 1, 6caovcom 6796 . . 3 (𝐶𝐹𝐴) = (𝐴𝐹𝐶)
85, 2, 6caovcom 6796 . . 3 (𝐶𝐹𝐵) = (𝐵𝐹𝐶)
97, 8breq12i 4632 . 2 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))
104, 9syl6bb 276 1 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  Vcvv 3190   class class class wbr 4623  (class class class)co 6615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618
This theorem is referenced by:  caovord3  6812  genpnmax  9789  addclprlem1  9798  mulclprlem  9801  distrlem4pr  9808  ltexprlem6  9823  reclem3pr  9831  ltsosr  9875
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