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Theorem iscomlaw 42151
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 476 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 oveq 6696 . . . . . 6 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
3 oveq 6696 . . . . . 6 (𝑜 = → (𝑦𝑜𝑥) = (𝑦 𝑥))
42, 3eqeq12d 2666 . . . . 5 (𝑜 = → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 𝑦) = (𝑦 𝑥)))
54adantr 480 . . . 4 ((𝑜 = 𝑚 = 𝑀) → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 𝑦) = (𝑦 𝑥)))
61, 5raleqbidv 3182 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
71, 6raleqbidv 3182 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
8 df-comlaw 42148 . 2 comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
97, 8brabga 5018 1 (( 𝑉𝑀𝑊) → ( comLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941   class class class wbr 4685  (class class class)co 6690   comLaw ccomlaw 42146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-iota 5889  df-fv 5934  df-ov 6693  df-comlaw 42148
This theorem is referenced by: (None)
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