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Theorem 3jaoian 1211
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 112 . . 3 (𝜑 → (𝜓𝜒))
3 3jaoian.2 . . . 4 ((𝜃𝜓) → 𝜒)
43ex 112 . . 3 (𝜃 → (𝜓𝜒))
5 3jaoian.3 . . . 4 ((𝜏𝜓) → 𝜒)
65ex 112 . . 3 (𝜏 → (𝜓𝜒))
72, 4, 63jaoi 1209 . 2 ((𝜑𝜃𝜏) → (𝜓𝜒))
87imp 119 1 (((𝜑𝜃𝜏) ∧ 𝜓) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  w3o 895
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898
This theorem is referenced by:  xrltnsym  8815  xrlttr  8817  xltnegi  8849  qbtwnxr  9214
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