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Theorem caovordid 6019
Description: Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.)
Hypotheses
Ref Expression
caovordig.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
caovordid.2 (𝜑𝐴𝑆)
caovordid.3 (𝜑𝐵𝑆)
caovordid.4 (𝜑𝐶𝑆)
Assertion
Ref Expression
caovordid (𝜑 → (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 caovordid
StepHypRef Expression
1 id 19 . 2 (𝜑𝜑)
2 caovordid.2 . 2 (𝜑𝐴𝑆)
3 caovordid.3 . 2 (𝜑𝐵𝑆)
4 caovordid.4 . 2 (𝜑𝐶𝑆)
5 caovordig.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦)))
65caovordig 6018 . 2 ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
71, 2, 3, 4, 6syl13anc 1235 1 (𝜑 → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973  wcel 2141   class class class wbr 3989  (class class class)co 5853
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856
This theorem is referenced by: (None)
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