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Theorem asslawass 42414
Description: Associativity of an associative operation. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 21-Jan-2020.)
Assertion
Ref Expression
asslawass ( assLaw 𝑀 → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
Distinct variable groups:   𝑥,𝑀,𝑦,𝑧   𝑥, ,𝑦,𝑧

Proof of Theorem asslawass
Dummy variables 𝑚 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-asslaw 42409 . . . 4 assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
21bropaex12 5407 . . 3 ( assLaw 𝑀 → ( ∈ V ∧ 𝑀 ∈ V))
3 isasslaw 42413 . . 3 (( ∈ V ∧ 𝑀 ∈ V) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
42, 3syl 17 . 2 ( assLaw 𝑀 → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
54ibi 258 1 ( assLaw 𝑀 → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wral 3107  Vcvv 3402   class class class wbr 4855  (class class class)co 6883   assLaw casslaw 42405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-xp 5330  df-iota 6073  df-fv 6118  df-ov 6886  df-asslaw 42409
This theorem is referenced by: (None)
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