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Theorem iscomlaw 41614
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 475 . . 3 ((𝑜 = 𝑚 = 𝑀) → 𝑚 = 𝑀)
2 oveq 6529 . . . . . 6 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
3 oveq 6529 . . . . . 6 (𝑜 = → (𝑦𝑜𝑥) = (𝑦 𝑥))
42, 3eqeq12d 2620 . . . . 5 (𝑜 = → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 𝑦) = (𝑦 𝑥)))
54adantr 479 . . . 4 ((𝑜 = 𝑚 = 𝑀) → ((𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ (𝑥 𝑦) = (𝑦 𝑥)))
61, 5raleqbidv 3124 . . 3 ((𝑜 = 𝑚 = 𝑀) → (∀𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
71, 6raleqbidv 3124 . 2 ((𝑜 = 𝑚 = 𝑀) → (∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥) ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
8 df-comlaw 41611 . 2 comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
97, 8brabga 4900 1 (( 𝑉𝑀𝑊) → ( comLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  wral 2891   class class class wbr 4573  (class class class)co 6523   comLaw ccomlaw 41609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-iota 5750  df-fv 5794  df-ov 6526  df-comlaw 41611
This theorem is referenced by: (None)
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