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Theorem 2albiim 1898
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 1897 . . 3 (∀𝑦(𝜑𝜓) ↔ (∀𝑦(𝜑𝜓) ∧ ∀𝑦(𝜓𝜑)))
21albii 1827 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(∀𝑦(𝜑𝜓) ∧ ∀𝑦(𝜓𝜑)))
3 19.26 1878 . 2 (∀𝑥(∀𝑦(𝜑𝜓) ∧ ∀𝑦(𝜓𝜑)) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))
42, 3bitri 278 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817
This theorem depends on definitions:  df-bi 210  df-an 400
This theorem is referenced by:  sbnf2  2358  2eu6  2658  eqopab2bw  5444  eqopab2b  5448  eqrel  5670  eqrelrel  5682  eqoprab2bw  7300  eqoprab2b  7301  eqrelrd2  30702  eqrel2  36202  relcnveq2  36225  elrelscnveq2  36378  pm14.123a  41749
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