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Theorem 3jcad 1178
Description: Deduction conjoining the consequents of three implications. (Contributed by NM, 25-Sep-2005.)
Hypotheses
Ref Expression
3jcad.1 (𝜑 → (𝜓𝜒))
3jcad.2 (𝜑 → (𝜓𝜃))
3jcad.3 (𝜑 → (𝜓𝜏))
Assertion
Ref Expression
3jcad (𝜑 → (𝜓 → (𝜒𝜃𝜏)))

Proof of Theorem 3jcad
StepHypRef Expression
1 3jcad.1 . . . 4 (𝜑 → (𝜓𝜒))
21imp 124 . . 3 ((𝜑𝜓) → 𝜒)
3 3jcad.2 . . . 4 (𝜑 → (𝜓𝜃))
43imp 124 . . 3 ((𝜑𝜓) → 𝜃)
5 3jcad.3 . . . 4 (𝜑 → (𝜓𝜏))
65imp 124 . . 3 ((𝜑𝜓) → 𝜏)
72, 4, 63jca 1177 . 2 ((𝜑𝜓) → (𝜒𝜃𝜏))
87ex 115 1 (𝜑 → (𝜓 → (𝜒𝜃𝜏)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108
This theorem depends on definitions:  df-bi 117  df-3an 980
This theorem is referenced by:  ixxssixx  9871  iccid  9894  fzen  10011
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