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Theorem aaan 2206
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 2134 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑦𝜓))
32albii 1787 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 ∧ ∀𝑦𝜓))
4 aaan.2 . . . 4 𝑥𝜓
54nfal 2191 . . 3 𝑥𝑦𝜓
6519.27 2133 . 2 (∀𝑥(𝜑 ∧ ∀𝑦𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
73, 6bitri 264 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wal 1521  wnf 1748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-11 2074  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745  df-nf 1750
This theorem is referenced by:  bj-mo3OLD  32957  aaanv  38905  pm11.71  38914
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