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Theorem mpjao3dan 1202
Description: Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.)
Hypotheses
Ref Expression
mpjao3dan.1 ((𝜑𝜓) → 𝜒)
mpjao3dan.2 ((𝜑𝜃) → 𝜒)
mpjao3dan.3 ((𝜑𝜏) → 𝜒)
mpjao3dan.4 (𝜑 → (𝜓𝜃𝜏))
Assertion
Ref Expression
mpjao3dan (𝜑𝜒)

Proof of Theorem mpjao3dan
StepHypRef Expression
1 mpjao3dan.1 . . 3 ((𝜑𝜓) → 𝜒)
2 mpjao3dan.2 . . 3 ((𝜑𝜃) → 𝜒)
31, 2jaodan 710 . 2 ((𝜑 ∧ (𝜓𝜃)) → 𝜒)
4 mpjao3dan.3 . 2 ((𝜑𝜏) → 𝜒)
5 mpjao3dan.4 . . 3 (𝜑 → (𝜓𝜃𝜏))
6 df-3or 886 . . 3 ((𝜓𝜃𝜏) ↔ ((𝜓𝜃) ∨ 𝜏))
75, 6sylib 127 . 2 (𝜑 → ((𝜓𝜃) ∨ 𝜏))
83, 4, 7mpjaodan 711 1 (𝜑𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wo 629  w3o 884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630
This theorem depends on definitions:  df-bi 110  df-3or 886
This theorem is referenced by:  wetriext  4301  nntri3  6075  nntri2or2  6076  caucvgprlemnkj  6762  caucvgprlemnbj  6763  caucvgprprlemnkj  6788  caucvgprprlemnbj  6789  caucvgsr  6884
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