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Theorem 3impexpbicom 1343
Description: 3impexp 1342 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.)
Assertion
Ref Expression
3impexpbicom (((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))

Proof of Theorem 3impexpbicom
StepHypRef Expression
1 bicom 132 . . . 4 ((𝜃𝜏) ↔ (𝜏𝜃))
2 imbi2 230 . . . . 5 (((𝜃𝜏) ↔ (𝜏𝜃)) → (((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ ((𝜑𝜓𝜒) → (𝜏𝜃))))
32biimpcd 152 . . . 4 (((𝜑𝜓𝜒) → (𝜃𝜏)) → (((𝜃𝜏) ↔ (𝜏𝜃)) → ((𝜑𝜓𝜒) → (𝜏𝜃))))
41, 3mpi 15 . . 3 (((𝜑𝜓𝜒) → (𝜃𝜏)) → ((𝜑𝜓𝜒) → (𝜏𝜃)))
543expd 1132 . 2 (((𝜑𝜓𝜒) → (𝜃𝜏)) → (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))
6 3impexp 1342 . . . 4 (((𝜑𝜓𝜒) → (𝜏𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))
76biimpri 128 . . 3 ((𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))) → ((𝜑𝜓𝜒) → (𝜏𝜃)))
87, 1syl6ibr 155 . 2 ((𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))) → ((𝜑𝜓𝜒) → (𝜃𝜏)))
95, 8impbii 121 1 (((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  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
This theorem depends on definitions:  df-bi 114  df-3an 898
This theorem is referenced by: (None)
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