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Theorem 2albiim 1814
Description: Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005.)
Assertion
Ref Expression
2albiim (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))

Proof of Theorem 2albiim
StepHypRef Expression
1 albiim 1813 . . 3 (∀𝑦(𝜑𝜓) ↔ (∀𝑦(𝜑𝜓) ∧ ∀𝑦(𝜓𝜑)))
21albii 1744 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(∀𝑦(𝜑𝜓) ∧ ∀𝑦(𝜓𝜑)))
3 19.26 1795 . 2 (∀𝑥(∀𝑦(𝜑𝜓) ∧ ∀𝑦(𝜓𝜑)) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))
42, 3bitri 264 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734
This theorem depends on definitions:  df-bi 197  df-an 386
This theorem is referenced by:  sbnf2  2438  2eu6  2557  eqopab2b  4975  eqrel  5180  eqrelrel  5192  eqoprab2b  6678  eqrelrd2  29310  pm14.123a  38147
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