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

Theorem 3jao 1542
Description: Disjunction of three antecedents. (Contributed by NM, 8-Apr-1994.)
Assertion
Ref Expression
3jao (((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)) → ((𝜑𝜒𝜃) → 𝜓))

Proof of Theorem 3jao
StepHypRef Expression
1 jao 974 . . 3 ((𝜑𝜓) → ((𝜒𝜓) → ((𝜑𝜒) → 𝜓)))
2 df-3or 1101 . . . 4 ((𝜑𝜒𝜃) ↔ ((𝜑𝜒) ∨ 𝜃))
3 jao 974 . . . 4 (((𝜑𝜒) → 𝜓) → ((𝜃𝜓) → (((𝜑𝜒) ∨ 𝜃) → 𝜓)))
42, 3syl7bi 246 . . 3 (((𝜑𝜒) → 𝜓) → ((𝜃𝜓) → ((𝜑𝜒𝜃) → 𝜓)))
51, 4syl6 35 . 2 ((𝜑𝜓) → ((𝜒𝜓) → ((𝜃𝜓) → ((𝜑𝜒𝜃) → 𝜓))))
653imp 1130 1 (((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)) → ((𝜑𝜒𝜃) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 865  w3o 1099  w3a 1100
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 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102
This theorem is referenced by:  3jaob  1544  3jaoi  1545  3jaod  1546
  Copyright terms: Public domain W3C validator