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Theorem 19.35-1 1561
Description: Forward direction of Theorem 19.35 of [Margaris] p. 90. The converse holds for classical logic but not (for all propositions) in intuitionistic logic (Contributed by Mario Carneiro, 2-Feb-2015.)
Assertion
Ref Expression
19.35-1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem 19.35-1
StepHypRef Expression
1 19.29 1557 . . 3 ((∀𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∃𝑥(𝜑 ∧ (𝜑𝜓)))
2 pm3.35 340 . . . 4 ((𝜑 ∧ (𝜑𝜓)) → 𝜓)
32eximi 1537 . . 3 (∃𝑥(𝜑 ∧ (𝜑𝜓)) → ∃𝑥𝜓)
41, 3syl 14 . 2 ((∀𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∃𝑥𝜓)
54expcom 115 1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1288  wex 1427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-ial 1473
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.35i  1562  19.25  1563  19.36-1  1609  19.37-1  1610  spimt  1672  sbequi  1768
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