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Theorem pm11.62 37410
Description: Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.62 (∀𝑥𝑦((𝜑𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓𝜒)))
Distinct variable group:   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem pm11.62
StepHypRef Expression
1 impexp 461 . . . 4 (((𝜑𝜓) → 𝜒) ↔ (𝜑 → (𝜓𝜒)))
21albii 1737 . . 3 (∀𝑦((𝜑𝜓) → 𝜒) ↔ ∀𝑦(𝜑 → (𝜓𝜒)))
3 19.21v 1855 . . 3 (∀𝑦(𝜑 → (𝜓𝜒)) ↔ (𝜑 → ∀𝑦(𝜓𝜒)))
42, 3bitri 263 . 2 (∀𝑦((𝜑𝜓) → 𝜒) ↔ (𝜑 → ∀𝑦(𝜓𝜒)))
54albii 1737 1 (∀𝑥𝑦((𝜑𝜓) → 𝜒) ↔ ∀𝑥(𝜑 → ∀𝑦(𝜓𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  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
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696
This theorem is referenced by: (None)
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