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Theorem 19.36v 1983
Description: Version of 19.36 2215 with a disjoint variable condition instead of a nonfreeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)
Assertion
Ref Expression
19.36v (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.36v
StepHypRef Expression
1 19.35 1872 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.9v 1979 . . 3 (∃𝑥𝜓𝜓)
32imbi2i 336 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
41, 3bitri 275 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963
This theorem depends on definitions:  df-bi 206  df-ex 1774
This theorem is referenced by:  19.12vvv  1984  19.12vv  2337  ax13lem2  2369  axexte  2698  spcimdv  3577  bnj1090  34519  bj-spimvwt  36054  bj-spcimdv  36282  bj-spcimdvv  36283  19.36vv  43699
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