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Theorem 19.37-1 1684
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 1564 . 2 (∀𝑥𝜑𝜑)
3 19.35-1 1634 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
42, 3biimtrrid 153 1 (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1361  wnf 1470  wex 1502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-ial 1544
This theorem depends on definitions:  df-bi 117  df-nf 1471
This theorem is referenced by:  19.37aiv  1685  spcimegft  2827  eqvincg  2873
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