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Theorem 3jaodan 1242
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 113 . . 3 (𝜑 → (𝜓𝜒))
3 3jaodan.2 . . . 4 ((𝜑𝜃) → 𝜒)
43ex 113 . . 3 (𝜑 → (𝜃𝜒))
5 3jaodan.3 . . . 4 ((𝜑𝜏) → 𝜒)
65ex 113 . . 3 (𝜑 → (𝜏𝜒))
72, 4, 63jaod 1240 . 2 (𝜑 → ((𝜓𝜃𝜏) → 𝜒))
87imp 122 1 ((𝜑 ∧ (𝜓𝜃𝜏)) → 𝜒)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3o 923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926
This theorem is referenced by:  zeo  8821  xrltnsym  9232  xrlttr  9234  xrltso  9235  xrlttri3  9236  xltnegi  9266  qbtwnxr  9634
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