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Theorem 19.36-1 1604
Description: Closed form of 19.36i 1603. 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 1556 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.36-1.1 . . 3 𝑥𝜓
3219.9 1576 . 2 (∃𝑥𝜓𝜓)
41, 3syl6ib 159 1 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390  wex 1422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  vtocl2  2663  vtocl3  2664  spcimgft  2683
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