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Theorem 19.36 2223
Description: Theorem 19.36 of [Margaris] p. 90. See 19.36v 1991 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
Hypothesis
Ref Expression
19.36.1 𝑥𝜓
Assertion
Ref Expression
19.36 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Proof of Theorem 19.36
StepHypRef Expression
1 19.35 1880 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.36.1 . . . 4 𝑥𝜓
3219.9 2198 . . 3 (∃𝑥𝜓𝜓)
43imbi2i 335 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
51, 4bitri 274 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wex 1781  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171
This theorem depends on definitions:  df-bi 206  df-ex 1782  df-nf 1786
This theorem is referenced by:  19.36i  2224  19.12vv  2343  spcimgft  3577  19.12b  34768
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