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Theorem 2ralbiim 42839
 Description: Split a biconditional and distribute 2 quantifiers, analogous to 2albiim 1872 and ralbiim 3141. (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 3141 . . 3 (∀𝑦𝐵 (𝜑𝜓) ↔ (∀𝑦𝐵 (𝜑𝜓) ∧ ∀𝑦𝐵 (𝜓𝜑)))
21ralbii 3132 . 2 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ ∀𝑥𝐴 (∀𝑦𝐵 (𝜑𝜓) ∧ ∀𝑦𝐵 (𝜓𝜑)))
3 r19.26 3137 . 2 (∀𝑥𝐴 (∀𝑦𝐵 (𝜑𝜓) ∧ ∀𝑦𝐵 (𝜓𝜑)) ↔ (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∀𝑥𝐴𝑦𝐵 (𝜓𝜑)))
42, 3bitri 276 1 (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ↔ (∀𝑥𝐴𝑦𝐵 (𝜑𝜓) ∧ ∀𝑥𝐴𝑦𝐵 (𝜓𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∀wral 3105 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791 This theorem depends on definitions:  df-bi 208  df-an 397  df-ral 3110 This theorem is referenced by: (None)
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