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Theorem 19.37 2087
 Description: Theorem 19.37 of [Margaris] p. 90. See 19.37v 1897 for a version requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
Hypothesis
Ref Expression
19.37.1 𝑥𝜑
Assertion
Ref Expression
19.37 (∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Proof of Theorem 19.37
StepHypRef Expression
1 19.35 1794 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.37.1 . . . 4 𝑥𝜑
3219.3 2057 . . 3 (∀𝑥𝜑𝜑)
43imbi1i 338 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓))
51, 4bitri 263 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701 This theorem is referenced by:  bnj900  30253
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