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Theorem 19.36v 1852
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 1771 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.9v 1844 . . 3 (∃𝑥𝜓𝜓)
32imbi2i 321 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
41, 3bitri 259 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 191  wal 1466  wex 1692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836
This theorem depends on definitions:  df-bi 192  df-ex 1693
This theorem is referenced by:  19.36iv  1853  19.12vvv  1855  19.12vv  2124  ax13lem2  2180  axc9lem2OLD  2181  axext2  2486  vtocl2  3123  vtocl3  3124  bnj1090  29940  bj-spcimdv  31677  19.36vv  37089
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