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Theorem aaanv 39107
Description: Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 2330. (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 1994 . . 3 𝑦𝜑
2 nfv 1994 . . 3 𝑥𝜓
31, 2aaan 2330 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑦𝜓))
43bicomi 214 1 ((∀𝑥𝜑 ∧ ∀𝑦𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wal 1628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-10 2173  ax-11 2189  ax-12 2202
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-nf 1857
This theorem is referenced by: (None)
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