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Theorem caovassd 6047
Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovassg.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
caovassd.2 (𝜑𝐴𝑆)
caovassd.3 (𝜑𝐵𝑆)
caovassd.4 (𝜑𝐶𝑆)
Assertion
Ref Expression
caovassd (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovassd
StepHypRef Expression
1 id 19 . 2 (𝜑𝜑)
2 caovassd.2 . 2 (𝜑𝐴𝑆)
3 caovassd.3 . 2 (𝜑𝐵𝑆)
4 caovassd.4 . 2 (𝜑𝐶𝑆)
5 caovassg.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
65caovassg 6046 . 2 ((𝜑 ∧ (𝐴𝑆𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
71, 2, 3, 4, 6syl13anc 1250 1 (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 979   = wceq 1363  wcel 2158  (class class class)co 5888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891
This theorem is referenced by:  caov32d  6068  caov12d  6069  caov13d  6071  caov4d  6072  caovdilemd  6079  caovimo  6081  enq0tr  7446  prarloclemlo  7506  ltsosr  7776  grpinvalem  12822  grpinva  12823  grprida  12824  grprcan  12933
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