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Theorem 19.37-1 1605
Description: One direction of Theorem 19.37 of [Margaris] p. 90. The converse holds in classical logic but not, in general, here. (Contributed by Jim Kingdon, 21-Jun-2018.)
Hypothesis
Ref Expression
19.37-1.1 𝑥𝜑
Assertion
Ref Expression
19.37-1 (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))

Proof of Theorem 19.37-1
StepHypRef Expression
1 19.37-1.1 . . 3 𝑥𝜑
2119.3 1487 . 2 (∀𝑥𝜑𝜑)
3 19.35-1 1556 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
42, 3syl5bir 151 1 (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))
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:  19.37aiv  1606  spcimegft  2687  eqvincg  2729
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