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Theorem 3impexpbicom 1324
Description: 3impexp 1323 with biconditional consequent of antecedent that is commuted in consequent. (Contributed by Alan Sare, 31-Dec-2011.)
Assertion
Ref Expression
3impexpbicom (((φ ψ χ) → (θτ)) ↔ (φ → (ψ → (χ → (τθ)))))

Proof of Theorem 3impexpbicom
StepHypRef Expression
1 bicom 128 . . . 4 ((θτ) ↔ (τθ))
2 imbi2 226 . . . . 5 (((θτ) ↔ (τθ)) → (((φ ψ χ) → (θτ)) ↔ ((φ ψ χ) → (τθ))))
32biimpcd 148 . . . 4 (((φ ψ χ) → (θτ)) → (((θτ) ↔ (τθ)) → ((φ ψ χ) → (τθ))))
41, 3mpi 15 . . 3 (((φ ψ χ) → (θτ)) → ((φ ψ χ) → (τθ)))
543expd 1120 . 2 (((φ ψ χ) → (θτ)) → (φ → (ψ → (χ → (τθ)))))
6 3impexp 1323 . . . 4 (((φ ψ χ) → (τθ)) ↔ (φ → (ψ → (χ → (τθ)))))
76biimpri 124 . . 3 ((φ → (ψ → (χ → (τθ)))) → ((φ ψ χ) → (τθ)))
87, 1syl6ibr 151 . 2 ((φ → (ψ → (χ → (τθ)))) → ((φ ψ χ) → (θτ)))
95, 8impbii 117 1 (((φ ψ χ) → (θτ)) ↔ (φ → (ψ → (χ → (τθ)))))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   w3a 884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101
This theorem depends on definitions:  df-bi 110  df-3an 886
This theorem is referenced by: (None)
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