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Theorem 19.31vv 37429
Description: Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
19.31vv (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑𝜓))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.31vv
StepHypRef Expression
1 19.31v 1856 . . 3 (∀𝑦(𝜑𝜓) ↔ (∀𝑦𝜑𝜓))
21albii 1736 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(∀𝑦𝜑𝜓))
3 19.31v 1856 . 2 (∀𝑥(∀𝑦𝜑𝜓) ↔ (∀𝑥𝑦𝜑𝜓))
42, 3bitri 262 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wo 381  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826
This theorem depends on definitions:  df-bi 195  df-or 383  df-ex 1695
This theorem is referenced by: (None)
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