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Theorem caovcomd 6006
Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovcomg.1 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovcomd.2 (𝜑𝐴𝑆)
caovcomd.3 (𝜑𝐵𝑆)
Assertion
Ref Expression
caovcomd (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦

Proof of Theorem caovcomd
StepHypRef Expression
1 id 19 . 2 (𝜑𝜑)
2 caovcomd.2 . 2 (𝜑𝐴𝑆)
3 caovcomd.3 . 2 (𝜑𝐵𝑆)
4 caovcomg.1 . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
54caovcomg 6005 . 2 ((𝜑 ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
61, 2, 3, 5syl12anc 1231 1 (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  (class class class)co 5850
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 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-iota 5158  df-fv 5204  df-ov 5853
This theorem is referenced by:  caovcanrd  6013  caovord2d  6019  caovdir2d  6026  caov32d  6030  caov12d  6031  caov31d  6032  caov411d  6035  caov42d  6036  caovimo  6043  ecopovsymg  6608  ecopoverg  6610  ltsonq  7347  prarloclemlo  7443  addextpr  7570  ltsosr  7713  ltasrg  7719  mulextsr1lem  7729  seq3f1olemqsumkj  10441
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