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Theorem iscomlaw 44511
 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 488 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 oveq 7146 . . . . . 6 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
3 oveq 7146 . . . . . 6 (𝑜 = → (𝑦𝑜𝑥) = (𝑦 𝑥))
42, 3eqeq12d 2814 . . . . 5 (𝑜 = → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 𝑦) = (𝑦 𝑥)))
54adantr 484 . . . 4 ((𝑜 = 𝑚 = 𝑀) → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 𝑦) = (𝑦 𝑥)))
61, 5raleqbidv 3354 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
71, 6raleqbidv 3354 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
8 df-comlaw 44508 . 2 comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
97, 8brabga 5387 1 (( 𝑉𝑀𝑊) → ( comLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   class class class wbr 5031  (class class class)co 7140   comLaw ccomlaw 44506 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-iota 6286  df-fv 6335  df-ov 7143  df-comlaw 44508 This theorem is referenced by: (None)
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