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Theorem 3jaoian 1284
 Description: Disjunction of 3 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 114 . . 3 (𝜑 → (𝜓𝜒))
3 3jaoian.2 . . . 4 ((𝜃𝜓) → 𝜒)
43ex 114 . . 3 (𝜃 → (𝜓𝜒))
5 3jaoian.3 . . . 4 ((𝜏𝜓) → 𝜒)
65ex 114 . . 3 (𝜏 → (𝜓𝜒))
72, 4, 63jaoi 1282 . 2 ((𝜑𝜃𝜏) → (𝜓𝜒))
87imp 123 1 (((𝜑𝜃𝜏) ∧ 𝜓) → 𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ w3o 962 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699 This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965 This theorem is referenced by:  xrltnsym  9638  xrlttr  9640  xltnegi  9677  xaddcom  9703  xnegdi  9710  qbtwnxr  10095
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