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Theorem ralbiim 3111
Description: Split a biconditional and distribute quantifier. Restricted quantifier version of albiim 1887. (Contributed by NM, 3-Jun-2012.)
Assertion
Ref Expression
ralbiim (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))

Proof of Theorem ralbiim
StepHypRef Expression
1 dfbi2 474 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21ralbii 3091 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ ∀𝑥𝐴 ((𝜑𝜓) ∧ (𝜓𝜑)))
3 r19.26 3109 . 2 (∀𝑥𝐴 ((𝜑𝜓) ∧ (𝜓𝜑)) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))
42, 3bitri 275 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 (𝜑𝜓) ∧ ∀𝑥𝐴 (𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wral 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806
This theorem depends on definitions:  df-bi 207  df-an 396  df-ral 3060
This theorem is referenced by:  2ralbiim  3130  eqreu  3738  isclo2  23112  chrelat4i  32402  hlateq  39382  ntrneik13  44088  ntrneix13  44089
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