MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  3jaoian Structured version   Visualization version   GIF version

Theorem 3jaoian 1555
Description: Disjunction of three 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 402 . . 3 (𝜑 → (𝜓𝜒))
3 3jaoian.2 . . . 4 ((𝜃𝜓) → 𝜒)
43ex 402 . . 3 (𝜃 → (𝜓𝜒))
5 3jaoian.3 . . . 4 ((𝜏𝜓) → 𝜒)
65ex 402 . . 3 (𝜏 → (𝜓𝜒))
72, 4, 63jaoi 1553 . 2 ((𝜑𝜃𝜏) → (𝜓𝜒))
87imp 396 1 (((𝜑𝜃𝜏) ∧ 𝜓) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3o 1107
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 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110
This theorem is referenced by:  xrltnsym  12213  xrlttri  12215  xrlttr  12216  qbtwnxr  12276  xltnegi  12292  xaddcom  12316  xnegdi  12323  lcmftp  15680  xaddeq0  30027  3ccased  32106
  Copyright terms: Public domain W3C validator