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Theorem 19.35 1794
 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 41 . . . 4 (𝜑 → ((𝜑𝜓) → 𝜓))
21aleximi 1749 . . 3 (∀𝑥𝜑 → (∃𝑥(𝜑𝜓) → ∃𝑥𝜓))
32com12 32 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
4 exnal 1744 . . . 4 (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
5 pm2.21 119 . . . . 5 𝜑 → (𝜑𝜓))
65eximi 1752 . . . 4 (∃𝑥 ¬ 𝜑 → ∃𝑥(𝜑𝜓))
74, 6sylbir 224 . . 3 (¬ ∀𝑥𝜑 → ∃𝑥(𝜑𝜓))
8 exa1 1756 . . 3 (∃𝑥𝜓 → ∃𝑥(𝜑𝜓))
97, 8ja 172 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
103, 9impbii 198 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀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 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by:  19.35i  1795  19.35ri  1796  19.25  1797  19.43  1799  nfimd  1812  speimfwALT  1864  19.39  1886  19.24  1887  19.36v  1891  19.37v  1897  19.36  2085  19.37  2087  spimt  2241  grothprim  9535  bj-nalnaleximiOLD  31798  bj-spimt2  31896  bj-spimtv  31905  bj-nfimt  32025  bj-snsetex  32144
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