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Theorem 19.37 2086
Description: Theorem 19.37 of [Margaris] p. 90. See 19.37v 1896 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 1793 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.37.1 . . . 4 𝑥𝜑
3219.3 2056 . . 3 (∀𝑥𝜑𝜑)
43imbi1i 337 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓))
51, 4bitri 262 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wal 1472  wex 1694  wnf 1698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-12 2033
This theorem depends on definitions:  df-bi 195  df-ex 1695  df-nf 1700
This theorem is referenced by:  bnj900  30087
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