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Theorem 2ralbiim 41694
Description: Split a biconditional and distribute 2 quantifiers, analogous to 2albiim 1969 and ralbiim 3217. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Assertion
Ref Expression
2ralbiim (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∀𝑥𝐴𝑦𝐵 (𝜓𝜑)))

Proof of Theorem 2ralbiim
StepHypRef Expression
1 ralbiim 3217 . . 3 (∀𝑦𝐵 (𝜑𝜓) ↔ (∀𝑦𝐵 (𝜑𝜓) ∧ ∀𝑦𝐵 (𝜓𝜑)))
21ralbii 3129 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (∀𝑦𝐵 (𝜑𝜓) ∧ ∀𝑦𝐵 (𝜓𝜑)))
3 r19.26 3212 . 2 (∀𝑥𝐴 (∀𝑦𝐵 (𝜑𝜓) ∧ ∀𝑦𝐵 (𝜓𝜑)) ↔ (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∀𝑥𝐴𝑦𝐵 (𝜓𝜑)))
42, 3bitri 264 1 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∀𝑥𝐴𝑦𝐵 (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wral 3061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885
This theorem depends on definitions:  df-bi 197  df-an 383  df-ral 3066
This theorem is referenced by: (None)
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