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Theorem albiim 1419
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 380 . . 3 ((𝜑𝜓) ↔ ((𝜑𝜓) ∧ (𝜓𝜑)))
21albii 1402 . 2 (∀𝑥(𝜑𝜓) ↔ ∀𝑥((𝜑𝜓) ∧ (𝜓𝜑)))
3 19.26 1413 . 2 (∀𝑥((𝜑𝜓) ∧ (𝜓𝜑)) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
42, 3bitri 182 1 (∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  2albiim  1420  hbbid  1510  equveli  1686  spsbbi  1769  eu1  1970  eqss  3029  ssext  4022
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