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Theorem 3impexpbicom 1367
 Description: 3impexp 1366 with biconditional consequent of antecedent that is commuted in consequent. Derived automatically from 3impexpVD in set.mm. (Contributed by Alan Sare, 31-Dec-2011.) (New usage is discouraged.) TODO: decide if this is worth keeping.
Assertion
Ref Expression
3impexpbicom (((φ ψ χ) → (θτ)) ↔ (φ → (ψ → (χ → (τθ)))))

Proof of Theorem 3impexpbicom
StepHypRef Expression
1 bicom 191 . . . 4 ((θτ) ↔ (τθ))
2 imbi2 314 . . . . 5 (((θτ) ↔ (τθ)) → (((φ ψ χ) → (θτ)) ↔ ((φ ψ χ) → (τθ))))
32biimpcd 215 . . . 4 (((φ ψ χ) → (θτ)) → (((θτ) ↔ (τθ)) → ((φ ψ χ) → (τθ))))
41, 3mpi 16 . . 3 (((φ ψ χ) → (θτ)) → ((φ ψ χ) → (τθ)))
543expd 1168 . 2 (((φ ψ χ) → (θτ)) → (φ → (ψ → (χ → (τθ)))))
6 3impexp 1366 . . . 4 (((φ ψ χ) → (τθ)) ↔ (φ → (ψ → (χ → (τθ)))))
76biimpri 197 . . 3 ((φ → (ψ → (χ → (τθ)))) → ((φ ψ χ) → (τθ)))
87, 1syl6ibr 218 . 2 ((φ → (ψ → (χ → (τθ)))) → ((φ ψ χ) → (θτ)))
95, 8impbii 180 1 (((φ ψ χ) → (θτ)) ↔ (φ → (ψ → (χ → (τθ)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934 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 177  df-an 360  df-3an 936 This theorem is referenced by: (None)
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