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Theorem 3impexpbicom 40793
 Description: Version of 3impexp 1352 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 224 . . . 4 ((𝜃𝜏) ↔ (𝜏𝜃))
2 imbi2 351 . . . . 5 (((𝜃𝜏) ↔ (𝜏𝜃)) → (((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ ((𝜑𝜓𝜒) → (𝜏𝜃))))
32biimpcd 251 . . . 4 (((𝜑𝜓𝜒) → (𝜃𝜏)) → (((𝜃𝜏) ↔ (𝜏𝜃)) → ((𝜑𝜓𝜒) → (𝜏𝜃))))
41, 3mpi 20 . . 3 (((𝜑𝜓𝜒) → (𝜃𝜏)) → ((𝜑𝜓𝜒) → (𝜏𝜃)))
543expd 1347 . 2 (((𝜑𝜓𝜒) → (𝜃𝜏)) → (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))
6 3impexp 1352 . . . 4 (((𝜑𝜓𝜒) → (𝜏𝜃)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))
76biimpri 230 . . 3 ((𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))) → ((𝜑𝜓𝜒) → (𝜏𝜃)))
87, 1syl6ibr 254 . 2 ((𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))) → ((𝜑𝜓𝜒) → (𝜃𝜏)))
95, 8impbii 211 1 (((𝜑𝜓𝜒) → (𝜃𝜏)) ↔ (𝜑 → (𝜓 → (𝜒 → (𝜏𝜃)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1081 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 209  df-an 399  df-3an 1083 This theorem is referenced by:  3impexpbicomiVD  41172
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