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Theorem 3jaodan 1212
 Description: Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)
Hypotheses
Ref Expression
3jaodan.1 ((𝜑𝜓) → 𝜒)
3jaodan.2 ((𝜑𝜃) → 𝜒)
3jaodan.3 ((𝜑𝜏) → 𝜒)
Assertion
Ref Expression
3jaodan ((𝜑 ∧ (𝜓𝜃𝜏)) → 𝜒)

Proof of Theorem 3jaodan
StepHypRef Expression
1 3jaodan.1 . . . 4 ((𝜑𝜓) → 𝜒)
21ex 112 . . 3 (𝜑 → (𝜓𝜒))
3 3jaodan.2 . . . 4 ((𝜑𝜃) → 𝜒)
43ex 112 . . 3 (𝜑 → (𝜃𝜒))
5 3jaodan.3 . . . 4 ((𝜑𝜏) → 𝜒)
65ex 112 . . 3 (𝜑 → (𝜏𝜒))
72, 4, 63jaod 1210 . 2 (𝜑 → ((𝜓𝜃𝜏) → 𝜒))
87imp 119 1 ((𝜑 ∧ (𝜓𝜃𝜏)) → 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:  zeo  8401  xrltnsym  8814  xrlttr  8816  xrltso  8817  xrlttri3  8818  xltnegi  8848  qbtwnxr  9213
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