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Theorem iscomlaw 46210
Description: The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
iscomlaw (( 𝑉𝑀𝑊) → ( comLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
Distinct variable groups:   𝑥,𝑀,𝑦   𝑥, ,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem iscomlaw
Dummy variables 𝑚 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 486 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 oveq 7364 . . . . . 6 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
3 oveq 7364 . . . . . 6 (𝑜 = → (𝑦𝑜𝑥) = (𝑦 𝑥))
42, 3eqeq12d 2749 . . . . 5 (𝑜 = → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 𝑦) = (𝑦 𝑥)))
54adantr 482 . . . 4 ((𝑜 = 𝑚 = 𝑀) → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 𝑦) = (𝑦 𝑥)))
61, 5raleqbidv 3318 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
71, 6raleqbidv 3318 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
8 df-comlaw 46207 . 2 comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
97, 8brabga 5492 1 (( 𝑉𝑀𝑊) → ( comLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061   class class class wbr 5106  (class class class)co 7358   comLaw ccomlaw 46205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-iota 6449  df-fv 6505  df-ov 7361  df-comlaw 46207
This theorem is referenced by: (None)
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