Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  3jaao GIF version

Theorem 3jaao 1240
 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 960 . 2 ((𝜑𝜃𝜂) → (𝜓𝜒))
3 3jaao.2 . . 3 (𝜃 → (𝜏𝜒))
433ad2ant2 961 . 2 ((𝜑𝜃𝜂) → (𝜏𝜒))
5 3jaao.3 . . 3 (𝜂 → (𝜁𝜒))
653ad2ant3 962 . 2 ((𝜑𝜃𝜂) → (𝜁𝜒))
72, 4, 63jaod 1236 1 ((𝜑𝜃𝜂) → ((𝜓𝜏𝜁) → 𝜒))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ w3o 919   ∧ w3a 920 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663 This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator