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Theorem 3bior1fd 1579
Description: A disjunction is equivalent to a threefold disjunction with single falsehood, analogous to biorf 419. (Contributed by Alexander van der Vekens, 8-Sep-2017.)
Hypothesis
Ref Expression
3biorfd.1 (𝜑 → ¬ 𝜃)
Assertion
Ref Expression
3bior1fd (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))

Proof of Theorem 3bior1fd
StepHypRef Expression
1 3biorfd.1 . . 3 (𝜑 → ¬ 𝜃)
2 biorf 419 . . 3 𝜃 → ((𝜒𝜓) ↔ (𝜃 ∨ (𝜒𝜓))))
31, 2syl 17 . 2 (𝜑 → ((𝜒𝜓) ↔ (𝜃 ∨ (𝜒𝜓))))
4 3orass 1075 . 2 ((𝜃𝜒𝜓) ↔ (𝜃 ∨ (𝜒𝜓)))
53, 4syl6bbr 278 1 (𝜑 → ((𝜒𝜓) ↔ (𝜃𝜒𝜓)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  w3o 1071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-or 384  df-3or 1073
This theorem is referenced by:  3bior1fand  1580  3bior2fd  1581  nb3grprlem2  26473
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