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Theorem albiim 1991
 Description: Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
Assertion
Ref Expression
albiim (∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))

Proof of Theorem albiim
StepHypRef Expression
1 dfbi2 468 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21albii 1918 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥((𝜑𝜓) ∧ (𝜓𝜑)))
3 19.26 1972 . 2 (∀𝑥((𝜑𝜓) ∧ (𝜓𝜑)) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
42, 3bitri 267 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1654 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908 This theorem depends on definitions:  df-bi 199  df-an 387 This theorem is referenced by:  2albiim  1992  mobi  2613  eu6lem  2644  dfmo  2668  eu1  2695  eu1OLD  2696  eqss  3842  ssext  5146  asymref2  5759  rabeqsnd  29886  pm14.122a  39461
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