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Theorem 19.45v 2081
Description: Version of 19.45 2263 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
Assertion
Ref Expression
19.45v (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Distinct variable group:   𝜑,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem 19.45v
StepHypRef Expression
1 19.43 1962 . 2 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2 19.9v 2065 . . 3 (∃𝑥𝜑𝜑)
32orbi1i 899 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
41, 3bitri 264 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 836  wex 1852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057
This theorem depends on definitions:  df-bi 197  df-or 837  df-ex 1853
This theorem is referenced by: (None)
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