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Theorem 3jaodan 1433
Description: Disjunction of three 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 412 . . 3 (𝜑 → (𝜓𝜒))
3 3jaodan.2 . . . 4 ((𝜑𝜃) → 𝜒)
43ex 412 . . 3 (𝜑 → (𝜃𝜒))
5 3jaodan.3 . . . 4 ((𝜑𝜏) → 𝜒)
65ex 412 . . 3 (𝜑 → (𝜏𝜒))
72, 4, 63jaod 1431 . 2 (𝜑 → ((𝜓𝜃𝜏) → 𝜒))
87imp 406 1 ((𝜑 ∧ (𝜓𝜃𝜏)) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3o 1086
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 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089
This theorem is referenced by:  mpjao3dan  1434  onzsl  7867  zeo  12704  xrltnsym  13179  xrlttri  13181  xrlttr  13182  qbtwnxr  13242  xltnegi  13258  xaddcom  13282  xnegdi  13290  xsubge0  13303  xrub  13354  bpoly3  16094  blssioo  24816  ismbf2d  25675  itg2seq  25777  eliccioo  32913  3ccased  35719  lineelsb2  36149  sticksstones1  42147  dfxlim2v  45862  usgrexmpl2trifr  47996
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