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Theorem 3jaoian 1426
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 416 . . 3 (𝜑 → (𝜓𝜒))
3 3jaoian.2 . . . 4 ((𝜃𝜓) → 𝜒)
43ex 416 . . 3 (𝜃 → (𝜓𝜒))
5 3jaoian.3 . . . 4 ((𝜏𝜓) → 𝜒)
65ex 416 . . 3 (𝜏 → (𝜓𝜒))
72, 4, 63jaoi 1424 . 2 ((𝜑𝜃𝜏) → (𝜓𝜒))
87imp 410 1 (((𝜑𝜃𝜏) ∧ 𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3o 1083
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 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086
This theorem is referenced by:  xrltnsym  12522  xrlttri  12524  xrlttr  12525  qbtwnxr  12585  xltnegi  12601  xaddcom  12625  xnegdi  12633  lcmftp  15973  xaddeq0  30506  3ccased  33057
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