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Theorem 19.36-1 1619
Description: Closed form of 19.36i 1618. One direction of Theorem 19.36 of [Margaris] p. 90. The converse holds in classical logic, but does not hold (for all propositions) in intuitionistic logic. (Contributed by Jim Kingdon, 20-Jun-2018.)
Hypothesis
Ref Expression
19.36-1.1 𝑥𝜓
Assertion
Ref Expression
19.36-1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))

Proof of Theorem 19.36-1
StepHypRef Expression
1 19.35-1 1571 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.36-1.1 . . 3 𝑥𝜓
3219.9 1591 . 2 (∃𝑥𝜓𝜓)
41, 3syl6ib 160 1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1297  wnf 1404  wex 1436
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 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-4 1455  ax-ial 1482
This theorem depends on definitions:  df-bi 116  df-nf 1405
This theorem is referenced by:  vtocl2  2696  vtocl3  2697  spcimgft  2717
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