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Theorem ralbiim 2464
Description: Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)
Assertion
Ref Expression
ralbiim (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))

Proof of Theorem ralbiim
StepHypRef Expression
1 dfbi2 374 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21ralbii 2347 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 ((𝜑𝜓) ∧ (𝜓𝜑)))
3 r19.26 2458 . 2 (∀𝑥𝐴 ((𝜑𝜓) ∧ (𝜓𝜑)) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))
42, 3bitri 177 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-4 1416  ax-17 1435
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-ral 2328
This theorem is referenced by:  eqreu  2756
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