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Theorem 19.35 1771
Description: Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
Assertion
Ref Expression
19.35 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem 19.35
StepHypRef Expression
1 pm2.27 40 . . . 4 (𝜑 → ((𝜑𝜓) → 𝜓))
21aleximi 1735 . . 3 (∀𝑥𝜑 → (∃𝑥(𝜑𝜓) → ∃𝑥𝜓))
32com12 32 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4 exnal 1730 . . . 4 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
5 pm2.21 113 . . . . 5 𝜑 → (𝜑𝜓))
65eximi 1738 . . . 4 (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑𝜓))
74, 6sylbir 220 . . 3 (¬ ∀𝑥𝜑 → ∃𝑥(𝜑𝜓))
8 exa1 1742 . . 3 (∃𝑥𝜓 → ∃𝑥(𝜑𝜓))
97, 8ja 168 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
103, 9impbii 194 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 191  wal 1466  wex 1692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711
This theorem depends on definitions:  df-bi 192  df-ex 1693
This theorem is referenced by:  19.35i  1772  19.35ri  1773  19.25  1774  19.43  1776  speimfwALT  1825  19.39  1847  19.24  1848  19.36v  1852  19.37v  1858  19.36  2096  19.37  2098  spimt  2144  grothprim  9344  bj-nalnaleximiOLD  31404  bj-spimt2  31497  bj-spimtv  31506  bj-snsetex  31743
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