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Theorem aaan 2355
Description: Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
Hypotheses
Ref Expression
aaan.1 𝑦𝜑
aaan.2 𝑥𝜓
Assertion
Ref Expression
aaan (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))

Proof of Theorem aaan
StepHypRef Expression
1 aaan.1 . . . 4 𝑦𝜑
2119.28 2232 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
32albii 1821 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4 aaan.2 . . . 4 𝑥𝜓
54nfal 2344 . . 3 𝑥𝑦𝜓
6519.27 2231 . 2 (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
73, 6bitri 278 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wal 1536  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-10 2146  ax-11 2162  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by:  aaanv  41016  pm11.71  41025
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