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Theorem pm11.61 38075
Description: Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
Assertion
Ref Expression
pm11.61 (∃𝑦𝑥(𝜑𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))
Distinct variable group:   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem pm11.61
StepHypRef Expression
1 19.12 2161 . 2 (∃𝑦𝑥(𝜑𝜓) → ∀𝑥𝑦(𝜑𝜓))
2 19.37v 1907 . . . 4 (∃𝑦(𝜑𝜓) ↔ (𝜑 → ∃𝑦𝜓))
32biimpi 206 . . 3 (∃𝑦(𝜑𝜓) → (𝜑 → ∃𝑦𝜓))
43alimi 1736 . 2 (∀𝑥𝑦(𝜑𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))
51, 4syl 17 1 (∃𝑦𝑥(𝜑𝜓) → ∀𝑥(𝜑 → ∃𝑦𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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