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Theorem 19.35-1 1603
 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 1599 . . 3 ((∀𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∃𝑥(𝜑 ∧ (𝜑𝜓)))
2 pm3.35 344 . . . 4 ((𝜑 ∧ (𝜑𝜓)) → 𝜓)
32eximi 1579 . . 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 1329  ∃wex 1468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514 This theorem depends on definitions:  df-bi 116 This theorem is referenced by:  19.35i  1604  19.25  1605  19.36-1  1651  19.37-1  1652  spimt  1714  sbequi  1811
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