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Theorem 2ralor 3294
Description: Distribute restricted universal quantification over "or". (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Wolf Lammen, 20-Nov-2024.)
Assertion
Ref Expression
2ralor (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑥)   𝐵(𝑦)

Proof of Theorem 2ralor
StepHypRef Expression
1 r19.32v 3268 . . . 4 (∀𝑦𝐵 (𝜑𝜓) ↔ (𝜑 ∨ ∀𝑦𝐵 𝜓))
2 orcom 867 . . . 4 ((𝜑 ∨ ∀𝑦𝐵 𝜓) ↔ (∀𝑦𝐵 𝜓𝜑))
31, 2bitri 274 . . 3 (∀𝑦𝐵 (𝜑𝜓) ↔ (∀𝑦𝐵 𝜓𝜑))
43ralbii 3091 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (∀𝑦𝐵 𝜓𝜑))
5 r19.32v 3268 . 2 (∀𝑥𝐴 (∀𝑦𝐵 𝜓𝜑) ↔ (∀𝑦𝐵 𝜓 ∨ ∀𝑥𝐴 𝜑))
6 orcom 867 . 2 ((∀𝑦𝐵 𝜓 ∨ ∀𝑥𝐴 𝜑) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
74, 5, 63bitri 297 1 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∨ ∀𝑦𝐵 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 844  wral 3064
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 1913
This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1783  df-ral 3069
This theorem is referenced by:  prmidl2  31602  ispridl2  36182
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