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Theorem aaanv 40597
Description: Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2344. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
aaanv ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem aaanv
StepHypRef Expression
1 nfv 1906 . . 3 𝑦𝜑
2 nfv 1906 . . 3 𝑥𝜓
31, 2aaan 2344 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
43bicomi 225 1 ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wal 1526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-nf 1776
This theorem is referenced by: (None)
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