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Theorem 19.45 2105
 Description: Theorem 19.45 of [Margaris] p. 90. See 19.45v 1911 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
Hypothesis
Ref Expression
19.45.1 𝑥𝜑
Assertion
Ref Expression
19.45 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Proof of Theorem 19.45
StepHypRef Expression
1 19.43 1808 . 2 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2 19.45.1 . . . 4 𝑥𝜑
3219.9 2070 . . 3 (∃𝑥𝜑𝜑)
43orbi1i 542 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
51, 4bitri 264 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∨ wo 383  ∃wex 1702  Ⅎwnf 1706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-12 2045 This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1703  df-nf 1708 This theorem is referenced by:  eeor  2169
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