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Theorem 19.28vv 37607
 Description: Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
19.28vv (∀𝑥𝑦(𝜓𝜑) ↔ (𝜓 ∧ ∀𝑥𝑦𝜑))
Distinct variable groups:   𝜓,𝑥   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 19.28vv
StepHypRef Expression
1 19.28v 1896 . . 3 (∀𝑦(𝜓𝜑) ↔ (𝜓 ∧ ∀𝑦𝜑))
21albii 1737 . 2 (∀𝑥𝑦(𝜓𝜑) ↔ ∀𝑥(𝜓 ∧ ∀𝑦𝜑))
3 19.28v 1896 . 2 (∀𝑥(𝜓 ∧ ∀𝑦𝜑) ↔ (𝜓 ∧ ∀𝑥𝑦𝜑))
42, 3bitri 263 1 (∀𝑥𝑦(𝜓𝜑) ↔ (𝜓 ∧ ∀𝑥𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∀wal 1473 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696 This theorem is referenced by: (None)
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