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Theorem 19.35-1 1531
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 1527 . . 3 ((∀𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∃𝑥(𝜑 ∧ (𝜑𝜓)))
2 pm3.35 333 . . . 4 ((𝜑 ∧ (𝜑𝜓)) → 𝜓)
32eximi 1507 . . 3 (∃𝑥(𝜑 ∧ (𝜑𝜓)) → ∃𝑥𝜓)
41, 3syl 14 . 2 ((∀𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∃𝑥𝜓)
54expcom 113 1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wex 1397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  19.35i  1532  19.25  1533  19.36-1  1579  19.37-1  1580  spimt  1640  sbequi  1736
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