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Theorem 2albiim 1807
 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 1806 . . 3 (∀𝑦(𝜑𝜓) ↔ (∀𝑦(𝜑𝜓) ∧ ∀𝑦(𝜓𝜑)))
21albii 1737 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(∀𝑦(𝜑𝜓) ∧ ∀𝑦(𝜓𝜑)))
3 19.26 1786 . 2 (∀𝑥(∀𝑦(𝜑𝜓) ∧ ∀𝑦(𝜓𝜑)) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))
42, 3bitri 263 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728 This theorem depends on definitions:  df-bi 196  df-an 385 This theorem is referenced by:  sbnf2  2427  2eu6  2546  eqopab2b  4930  eqrel  5132  eqrelrel  5144  eqoprab2b  6611  eqrelrd2  28806  pm14.123a  37648
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