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Theorem 19.36v 1960
 Description: Version of 19.36 2136 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 1845 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.9v 1953 . . 3 (∃𝑥𝜓𝜓)
32imbi2i 325 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
41, 3bitri 264 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1521  ∃wex 1744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945 This theorem depends on definitions:  df-bi 197  df-ex 1745 This theorem is referenced by:  19.36ivOLD  1961  19.12vvv  1963  19.12vv  2216  ax13lem2  2332  axext2  2632  vtocl2  3292  vtocl3  3293  bnj1090  31173  bj-spimvwt  32781  bj-spcimdv  33009  bj-spcimdvv  33010  19.36vv  38899
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