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Theorem 19.35-1 1635
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 1631 . . 3 ((∀𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∃𝑥(𝜑 ∧ (𝜑𝜓)))
2 pm3.35 347 . . . 4 ((𝜑 ∧ (𝜑𝜓)) → 𝜓)
32eximi 1611 . . 3 (∃𝑥(𝜑 ∧ (𝜑𝜓)) → ∃𝑥𝜓)
41, 3syl 14 . 2 ((∀𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∃𝑥𝜓)
54expcom 116 1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1362  wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545
This theorem depends on definitions:  df-bi 117
This theorem is referenced by:  19.35i  1636  19.25  1637  19.36-1  1684  19.37-1  1685  spimt  1747  sbequi  1850
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