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Theorem 19.36v 1944
Description: Version of 19.36 2162 with a disjoint variable condition instead of a non-freeness 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 1840 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.9v 1938 . . 3 (∃𝑥𝜓𝜓)
32imbi2i 328 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
41, 3bitri 267 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1505  wex 1742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928
This theorem depends on definitions:  df-bi 199  df-ex 1743
This theorem is referenced by:  19.12vvv  1945  19.12vv  2281  ax13lem2  2305  axext2  2745  vtocl2OLD  3475  spcimdv  3505  bnj1090  31925  bj-spimvwt  33541  bj-spcimdv  33733  bj-spcimdvv  33734  19.36vv  40160
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