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Theorem 3jao 1381
 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 df-3or 1032 . 2 ((𝜑𝜒𝜃) ↔ ((𝜑𝜒) ∨ 𝜃))
2 jao 533 . . . 4 ((𝜑𝜓) → ((𝜒𝜓) → ((𝜑𝜒) → 𝜓)))
3 jao 533 . . . 4 (((𝜑𝜒) → 𝜓) → ((𝜃𝜓) → (((𝜑𝜒) ∨ 𝜃) → 𝜓)))
42, 3syl6 34 . . 3 ((𝜑𝜓) → ((𝜒𝜓) → ((𝜃𝜓) → (((𝜑𝜒) ∨ 𝜃) → 𝜓))))
543imp 1249 . 2 (((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)) → (((𝜑𝜒) ∨ 𝜃) → 𝜓))
61, 5syl5bi 231 1 (((𝜑𝜓) ∧ (𝜒𝜓) ∧ (𝜃𝜓)) → ((𝜑𝜒𝜃) → 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∨ w3o 1030   ∧ w3a 1031 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 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033 This theorem is referenced by:  3jaob  1382  3jaoi  1383  3jaod  1384
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