Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm11.61 Structured version   Visualization version   GIF version

Theorem pm11.61 40602
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 2337 . 2 (∃𝑦𝑥(𝜑𝜓) → ∀𝑥𝑦(𝜑𝜓))
2 19.37v 1989 . . . 4 (∃𝑦(𝜑𝜓) ↔ (𝜑 → ∃𝑦𝜓))
32biimpi 217 . . 3 (∃𝑦(𝜑𝜓) → (𝜑 → ∃𝑦𝜓))
43alimi 1803 . 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 1526  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-10 2136  ax-11 2151  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-or 842  df-ex 1772  df-nf 1776
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator