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Theorem 19.45v 2095
Description: Version of 19.45 2274 with a disjoint variable 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 1982 . 2 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2 19.9v 2080 . . 3 (∃𝑥𝜑𝜑)
32orbi1i 938 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
41, 3bitri 267 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wo 874  wex 1875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072
This theorem depends on definitions:  df-bi 199  df-or 875  df-ex 1876
This theorem is referenced by: (None)
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