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Theorem pm11.61 37615
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 2150 . 2 (∃𝑦𝑥(𝜑𝜓) → ∀𝑥𝑦(𝜑𝜓))
2 19.37v 1897 . . . 4 (∃𝑦(𝜑𝜓) ↔ (𝜑 → ∃𝑦𝜓))
32biimpi 205 . . 3 (∃𝑦(𝜑𝜓) → (𝜑 → ∃𝑦𝜓))
43alimi 1730 . 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 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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