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Theorem 19.44 2239
 Description: Theorem 19.44 of [Margaris] p. 90. See 19.44v 1999 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
Hypothesis
Ref Expression
19.44.1 𝑥𝜓
Assertion
Ref Expression
19.44 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Proof of Theorem 19.44
StepHypRef Expression
1 19.43 1883 . 2 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2 19.44.1 . . . 4 𝑥𝜓
3219.9 2205 . . 3 (∃𝑥𝜓𝜓)
43orbi2i 909 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (∃𝑥𝜑𝜓))
51, 4bitri 277 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∨ wo 843  ∃wex 1780  Ⅎwnf 1784 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2177 This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1781  df-nf 1785 This theorem is referenced by:  eeor  2354
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