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Theorem 19.45 1694
Description: Theorem 19.45 of [Margaris] p. 90. (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 1639 . 2 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2 19.45.1 . . . 4 𝑥𝜑
3219.9 1655 . . 3 (∃𝑥𝜑𝜑)
43orbi1i 764 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
51, 4bitri 184 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff set class
Syntax hints:  wb 105  wo 709  wnf 1471  wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-nf 1472
This theorem is referenced by:  eeor  1706
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