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Theorem 19.35 1775
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 1737 . . 3 (∀𝑥𝜑 → (∃𝑥(𝜑𝜓) → ∃𝑥𝜓))
32com12 32 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4 exnal 1732 . . . 4 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
5 pm2.21 118 . . . . 5 𝜑 → (𝜑𝜓))
65eximi 1740 . . . 4 (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑𝜓))
74, 6sylbir 223 . . 3 (¬ ∀𝑥𝜑 → ∃𝑥(𝜑𝜓))
8 exa1 1744 . . 3 (∃𝑥𝜓 → ∃𝑥(𝜑𝜓))
97, 8ja 171 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
103, 9impbii 197 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wal 1472  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by:  19.35i  1776  19.35ri  1777  19.25  1778  19.43  1780  speimfwALT  1827  19.39  1849  19.24  1850  19.36v  1854  19.37v  1860  19.36  2093  19.37  2095  spimt  2144  grothprim  9410  bj-nalnaleximiOLD  31633  bj-spimt2  31731  bj-spimtv  31740  bj-snsetex  31976  wl-nfimd  32314
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