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Theorem caovord2 5955
 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 5954 . 2 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
5 caovord2.3 . . . 4 𝐶 ∈ V
6 caovord2.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
75, 1, 6caovcom 5940 . . 3 (𝐶𝐹𝐴) = (𝐴𝐹𝐶)
85, 2, 6caovcom 5940 . . 3 (𝐶𝐹𝐵) = (𝐵𝐹𝐶)
97, 8breq12i 3948 . 2 ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))
104, 9syl6bb 195 1 (𝐶𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332   ∈ wcel 2112  Vcvv 2691   class class class wbr 3939  (class class class)co 5786 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2123 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-un 3082  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-br 3940  df-iota 5100  df-fv 5143  df-ov 5789 This theorem is referenced by:  caovord3  5956
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