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Theorem 3impexpbicom 39002
Description: Version of 3impexp 1311 where in addition the consequent is commuted. (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 212 . . . 4 ((𝜃𝜏) ↔ (𝜏𝜃))
2 imbi2 337 . . . . 5 (((𝜃𝜏) ↔ (𝜏𝜃)) → (((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ ((𝜑𝜓𝜒) → (𝜏𝜃))))
32biimpcd 239 . . . 4 (((𝜑𝜓𝜒) → (𝜃𝜏)) → (((𝜃𝜏) ↔ (𝜏𝜃)) → ((𝜑𝜓𝜒) → (𝜏𝜃))))
41, 3mpi 20 . . 3 (((𝜑𝜓𝜒) → (𝜃𝜏)) → ((𝜑𝜓𝜒) → (𝜏𝜃)))
543expd 1306 . 2 (((𝜑𝜓𝜒) → (𝜃𝜏)) → (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))
6 3impexp 1311 . . . 4 (((𝜑𝜓𝜒) → (𝜏𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))
76biimpri 218 . . 3 ((𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))) → ((𝜑𝜓𝜒) → (𝜏𝜃)))
87, 1syl6ibr 242 . 2 ((𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))) → ((𝜑𝜓𝜒) → (𝜃𝜏)))
95, 8impbii 199 1 (((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054
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 197  df-an 385  df-3an 1056
This theorem is referenced by:  3impexpbicomiVD  39407
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