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Theorem caovcomd 5893
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 5892 . 2 ((𝜑 ∧ (𝐴𝑆𝐵𝑆)) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
61, 2, 3, 5syl12anc 1197 1 (𝜑 → (𝐴𝐹𝐵) = (𝐵𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  (class class class)co 5740
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-iota 5056  df-fv 5099  df-ov 5743
This theorem is referenced by:  caovcanrd  5900  caovord2d  5906  caovdir2d  5913  caov32d  5917  caov12d  5918  caov31d  5919  caov411d  5922  caov42d  5923  caovimo  5930  ecopovsymg  6494  ecopoverg  6496  ltsonq  7170  prarloclemlo  7266  addextpr  7393  ltsosr  7536  ltasrg  7542  mulextsr1lem  7552  seq3f1olemqsumkj  10222
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