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Theorem 19.44 2271
Description: Theorem 19.44 of [Margaris] p. 90. See 19.44v 2092 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 1981 . 2 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2 19.44.1 . . . 4 𝑥𝜓
3219.9 2237 . . 3 (∃𝑥𝜓𝜓)
43orbi2i 936 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (∃𝑥𝜑𝜓))
51, 4bitri 266 1 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wo 873  wex 1874  wnf 1878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-12 2211
This theorem depends on definitions:  df-bi 198  df-or 874  df-ex 1875  df-nf 1879
This theorem is referenced by:  eeor  2334
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