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Theorem 19.19 2084
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 2060 . 2 (∃𝑥𝜑𝜑)
3 exbi 1762 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
42, 3syl5bbr 273 1 (∀𝑥(𝜑𝜓) → (𝜑 ↔ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wal 1473  wex 1695  wnf 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: (None)
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