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Theorem 3jaao 1430
 Description: Inference conjoining and disjoining the antecedents of three implications. (Contributed by Jeff Hankins, 15-Aug-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.)
Hypotheses
Ref Expression
3jaao.1 (𝜑 → (𝜓𝜒))
3jaao.2 (𝜃 → (𝜏𝜒))
3jaao.3 (𝜂 → (𝜁𝜒))
Assertion
Ref Expression
3jaao ((𝜑𝜃𝜂) → ((𝜓𝜏𝜁) → 𝜒))

Proof of Theorem 3jaao
StepHypRef Expression
1 3jaao.1 . . 3 (𝜑 → (𝜓𝜒))
213ad2ant1 1130 . 2 ((𝜑𝜃𝜂) → (𝜓𝜒))
3 3jaao.2 . . 3 (𝜃 → (𝜏𝜒))
433ad2ant2 1131 . 2 ((𝜑𝜃𝜂) → (𝜏𝜒))
5 3jaao.3 . . 3 (𝜂 → (𝜁𝜒))
653ad2ant3 1132 . 2 ((𝜑𝜃𝜂) → (𝜁𝜒))
72, 4, 63jaod 1425 1 ((𝜑𝜃𝜂) → ((𝜓𝜏𝜁) → 𝜒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1083   ∧ w3a 1084 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 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086 This theorem is referenced by:  lpni  28266  3ornot23  41202
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