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Theorem 19.37-1 1653
 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 1534 . 2 (∀𝑥𝜑𝜑)
3 19.35-1 1604 . 2 (∃𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∃𝑥𝜓))
42, 3syl5bir 152 1 (∃𝑥(𝜑𝜓) → (𝜑 → ∃𝑥𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1330  Ⅎwnf 1437  ∃wex 1469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515 This theorem depends on definitions:  df-bi 116  df-nf 1438 This theorem is referenced by:  19.37aiv  1654  spcimegft  2768  eqvincg  2814
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