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Theorem 3jaodan 1431
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 414 . . 3 (𝜑 → (𝜓𝜒))
3 3jaodan.2 . . . 4 ((𝜑𝜃) → 𝜒)
43ex 414 . . 3 (𝜑 → (𝜃𝜒))
5 3jaodan.3 . . . 4 ((𝜑𝜏) → 𝜒)
65ex 414 . . 3 (𝜑 → (𝜏𝜒))
72, 4, 63jaod 1429 . 2 (𝜑 → ((𝜓𝜃𝜏) → 𝜒))
87imp 408 1 ((𝜑 ∧ (𝜓𝜃𝜏)) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3o 1087
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 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090
This theorem is referenced by:  mpjao3dan  1432  onzsl  7835  zeo  12648  xrltnsym  13116  xrlttri  13118  xrlttr  13119  qbtwnxr  13179  xltnegi  13195  xaddcom  13219  xnegdi  13227  xsubge0  13240  xrub  13291  bpoly3  16002  blssioo  24311  ismbf2d  25157  itg2seq  25260  eliccioo  32097  3ccased  34688  lineelsb2  35120  sticksstones1  40962  dfxlim2v  44563
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