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Theorem 19.37v 1912
Description: Version of 19.37 2103 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
Assertion
Ref Expression
19.37v (∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.37v
StepHypRef Expression
1 19.35 1805 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.3v 1899 . . 3 (∀𝑥𝜑𝜑)
32imbi1i 339 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓))
41, 3bitri 264 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))
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:  19.37iv  1913  axrep5  4741  fvn0ssdmfun  6307  kmlem14  8930  kmlem15  8931  eqvincg  29154  bnj132  30492  bnj1098  30554  bnj150  30646  bnj865  30693  bnj996  30725  bnj1021  30734  bnj1090  30747  bnj1176  30773  bj-axrep5  32427  cnvssco  37379  refimssco  37380  19.37vv  38052  pm11.61  38061  relopabVD  38606  rmoanim  40470
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