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Theorem i19.39 1574
 Description: Theorem 19.39 of [Margaris] p. 90, with an additional hypothesis. The hypothesis is the converse of 19.35-1 1558, and is a theorem of classical logic, but in intuitionistic logic it will only be provable for some propositions. (Contributed by Jim Kingdon, 22-Jul-2018.)
Hypothesis
Ref Expression
i19.24.1 ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
Assertion
Ref Expression
i19.39 ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Proof of Theorem i19.39
StepHypRef Expression
1 19.2 1572 . . 3 (∀𝑥𝜑 → ∃𝑥𝜑)
21imim1i 59 . 2 ((∃𝑥𝜑 → ∃𝑥𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
3 i19.24.1 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
42, 3syl 14 1 ((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1285  ∃wex 1424 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443 This theorem depends on definitions:  df-bi 115 This theorem is referenced by: (None)
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