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Theorem 3jaoian 1433
Description: Disjunction of three antecedents (inference). (Contributed by NM, 14-Oct-2005.)
Hypotheses
Ref Expression
3jaoian.1 ((𝜑𝜓) → 𝜒)
3jaoian.2 ((𝜃𝜓) → 𝜒)
3jaoian.3 ((𝜏𝜓) → 𝜒)
Assertion
Ref Expression
3jaoian (((𝜑𝜃𝜏) ∧ 𝜓) → 𝜒)

Proof of Theorem 3jaoian
StepHypRef Expression
1 3jaoian.1 . . . 4 ((𝜑𝜓) → 𝜒)
21ex 449 . . 3 (𝜑 → (𝜓𝜒))
3 3jaoian.2 . . . 4 ((𝜃𝜓) → 𝜒)
43ex 449 . . 3 (𝜃 → (𝜓𝜒))
5 3jaoian.3 . . . 4 ((𝜏𝜓) → 𝜒)
65ex 449 . . 3 (𝜏 → (𝜓𝜒))
72, 4, 63jaoi 1431 . 2 ((𝜑𝜃𝜏) → (𝜓𝜒))
87imp 444 1 (((𝜑𝜃𝜏) ∧ 𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3o 1053
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-an 385  df-3or 1055  df-3an 1056
This theorem is referenced by:  xrltnsym  12008  xrlttri  12010  xrlttr  12011  qbtwnxr  12069  xltnegi  12085  xaddcom  12109  xnegdi  12116  lcmftp  15396  xaddeq0  29646  3ccased  31726
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