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Theorem 3jaao 1214
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 936 . 2 ((𝜑𝜃𝜂) → (𝜓𝜒))
3 3jaao.2 . . 3 (𝜃 → (𝜏𝜒))
433ad2ant2 937 . 2 ((𝜑𝜃𝜂) → (𝜏𝜒))
5 3jaao.3 . . 3 (𝜂 → (𝜁𝜒))
653ad2ant3 938 . 2 ((𝜑𝜃𝜂) → (𝜁𝜒))
72, 4, 63jaod 1210 1 ((𝜑𝜃𝜂) → ((𝜓𝜏𝜁) → 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  w3o 895  w3a 896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898
This theorem is referenced by: (None)
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