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Theorem ralbiim 3144
Description: Split a biconditional and distribute quantifier. Restricted quantifier version of albiim 1890. (Contributed by NM, 3-Jun-2012.)
Assertion
Ref Expression
ralbiim (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))

Proof of Theorem ralbiim
StepHypRef Expression
1 dfbi2 478 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21ralbii 3136 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 ((𝜑𝜓) ∧ (𝜓𝜑)))
3 r19.26 3140 . 2 (∀𝑥𝐴 ((𝜑𝜓) ∧ (𝜓𝜑)) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))
42, 3bitri 278 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wral 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811
This theorem depends on definitions:  df-bi 210  df-an 400  df-ral 3114
This theorem is referenced by:  eqreu  3671  isclo2  21697  chrelat4i  30160  hlateq  36694  ntrneik13  40798  ntrneix13  40799  2ralbiim  43657
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