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Theorem albiim 1892
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 475 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21albii 1822 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥((𝜑𝜓) ∧ (𝜓𝜑)))
3 19.26 1873 . 2 (∀𝑥((𝜑𝜓) ∧ (𝜓𝜑)) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
42, 3bitri 274 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-an 397
This theorem is referenced by:  2albiim  1893  dfmoeu  2536  mobi  2547  eu6lem  2573  eu1  2612  eqss  3936  ssext  5369  asymref2  6016  rabeqsnd  30834  pm14.122a  41999
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