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Theorem 3jaoian 1390
 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 450 . . 3 (𝜑 → (𝜓𝜒))
3 3jaoian.2 . . . 4 ((𝜃𝜓) → 𝜒)
43ex 450 . . 3 (𝜃 → (𝜓𝜒))
5 3jaoian.3 . . . 4 ((𝜏𝜓) → 𝜒)
65ex 450 . . 3 (𝜏 → (𝜓𝜒))
72, 4, 63jaoi 1388 . 2 ((𝜑𝜃𝜏) → (𝜓𝜒))
87imp 445 1 (((𝜑𝜃𝜏) ∧ 𝜓) → 𝜒)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∨ w3o 1035 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 385  df-an 386  df-3or 1037  df-3an 1038 This theorem is referenced by:  xrltnsym  11930  xrlttri  11932  xrlttr  11933  qbtwnxr  11990  xltnegi  12006  xaddcom  12030  xnegdi  12037  lcmftp  15292  xaddeq0  29402  3ccased  31362
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