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Theorem 19.19 2081
Description: Theorem 19.19 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
Hypothesis
Ref Expression
19.19.1 𝑥𝜑
Assertion
Ref Expression
19.19 (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∃𝑥𝜓))

Proof of Theorem 19.19
StepHypRef Expression
1 19.19.1 . . 3 𝑥𝜑
2119.9 2057 . 2 (∃𝑥𝜑𝜑)
3 exbi 1760 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
42, 3syl5bbr 272 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 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-12 2031
This theorem depends on definitions:  df-bi 195  df-ex 1695  df-nf 1700
This theorem is referenced by: (None)
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