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Theorem caov32 5688
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
Hypotheses
Ref Expression
caov.1 𝐴 ∈ V
caov.2 𝐵 ∈ V
caov.3 𝐶 ∈ V
caov.com (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
caov.ass ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
Assertion
Ref Expression
caov32 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem caov32
StepHypRef Expression
1 caov.2 . . . 4 𝐵 ∈ V
2 caov.3 . . . 4 𝐶 ∈ V
3 caov.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
41, 2, 3caovcom 5658 . . 3 (𝐵𝐹𝐶) = (𝐶𝐹𝐵)
54oveq2i 5523 . 2 (𝐴𝐹(𝐵𝐹𝐶)) = (𝐴𝐹(𝐶𝐹𝐵))
6 caov.1 . . 3 𝐴 ∈ V
7 caov.ass . . 3 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
86, 1, 2, 7caovass 5661 . 2 ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
96, 2, 1, 7caovass 5661 . 2 ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵))
105, 8, 93eqtr4i 2070 1 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1243  wcel 1393  Vcvv 2557  (class class class)co 5512
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515
This theorem is referenced by:  caov31  5690
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