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Theorem pm11.53 2167
 Description: Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 1893 for a version requiring fewer axioms. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.53 (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem pm11.53
StepHypRef Expression
1 19.21v 1855 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
21albii 1737 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
3 nfv 1830 . . . 4 𝑥𝜓
43nfal 2139 . . 3 𝑥𝑦𝜓
5419.23 2067 . 2 (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
62, 5bitri 263 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695 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  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-ex 1696  df-nf 1701 This theorem is referenced by: (None)
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