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Theorem 19.36v 1906
 Description: Version of 19.36 2101 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 1805 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.9v 1898 . . 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 1478  ∃wex 1701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890 This theorem depends on definitions:  df-bi 197  df-ex 1702 This theorem is referenced by:  19.36iv  1907  19.12vvv  1909  19.12vv  2184  ax13lem2  2300  axext2  2607  vtocl2  3252  vtocl3  3253  bnj1090  30747  bj-spimvwt  32271  bj-spcimdv  32504  bj-spcimdvv  32505  19.36vv  38031
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