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Theorem 19.37v 1897
 Description: Version of 19.37 2087 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 1794 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.3v 1884 . . 3 (∀𝑥𝜑𝜑)
32imbi1i 338 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (𝜑 → ∃𝑥𝜓))
41, 3bitri 263 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by:  19.37iv  1898  axrep5  4704  fvn0ssdmfun  6258  kmlem14  8868  kmlem15  8869  eqvincg  28698  bnj132  30046  bnj1098  30108  bnj150  30200  bnj865  30247  bnj996  30279  bnj1021  30288  bnj1090  30301  bnj1176  30327  bj-axrep5  31980  cnvssco  36931  refimssco  36932  19.37vv  37606  pm11.61  37615  relopabVD  38159  rmoanim  39828
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