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Theorem 19.44 2104
Description: Theorem 19.44 of [Margaris] p. 90. See 19.44v 1910 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 1808 . 2 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2 19.44.1 . . . 4 𝑥𝜓
3219.9 2070 . . 3 (∃𝑥𝜓𝜓)
43orbi2i 541 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (∃𝑥𝜑𝜓))
51, 4bitri 264 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wex 1702  wnf 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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