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Theorem 19.36-1 1563
 Description: Closed form of 19.36i 1562. 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 1515 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.36-1.1 . . 3 𝑥𝜓
3219.9 1535 . 2 (∃𝑥𝜓𝜓)
41, 3syl6ib 150 1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241  Ⅎwnf 1349  ∃wex 1381 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  vtocl2  2609  vtocl3  2610  spcimgft  2629
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