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Theorem albiim 1446
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 383 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21albii 1429 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥((𝜑𝜓) ∧ (𝜓𝜑)))
3 19.26 1440 . 2 (∀𝑥((𝜑𝜓) ∧ (𝜓𝜑)) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
42, 3bitri 183 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  2albiim  1447  hbbid  1537  equveli  1715  spsbbi  1798  eu1  2000  eqss  3080  ssext  4111
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