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Theorem pm11.53 2343
Description: Theorem *11.53 in [WhiteheadRussell] p. 164. See pm11.53v 2039 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 2034 . . 3 (∀𝑦(𝜑𝜓) ↔ (𝜑 → ∀𝑦𝜓))
21albii 1914 . 2 (∀𝑥𝑦(𝜑𝜓) ↔ ∀𝑥(𝜑 → ∀𝑦𝜓))
3 nfv 2009 . . . 4 𝑥𝜓
43nfal 2326 . . 3 𝑥𝑦𝜓
5419.23 2244 . 2 (∀𝑥(𝜑 → ∀𝑦𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
62, 5bitri 266 1 (∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650  wex 1874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-10 2183  ax-11 2198  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-ex 1875  df-nf 1879
This theorem is referenced by: (None)
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