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Theorem 19.36v 1902
Description: Version of 19.36 2096 with a dv 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 1803 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.9v 1894 . . 3 (∃𝑥𝜓𝜓)
32imbi2i 326 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
41, 3bitri 264 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1479  wex 1702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886
This theorem depends on definitions:  df-bi 197  df-ex 1703
This theorem is referenced by:  19.36iv  1903  19.12vvv  1905  19.12vv  2178  ax13lem2  2294  axext2  2601  vtocl2  3256  vtocl3  3257  bnj1090  31021  bj-spimvwt  32631  bj-spcimdv  32859  bj-spcimdvv  32860  19.36vv  38402
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