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Proofgold Proof

pf
Let x0 of type ιιι be given.
Let x1 of type ιιι be given.
Assume H0: fb516.. x0.
Assume H1: fb516.. x1.
Apply unknownprop_fd3d2e7f783867b84b25b06a9614a22d40979688c105b0eabc60f25fb84d3565 with x0, x1, λ x2 x3 : ο . x3 = (x0 = x1) leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply prop_ext_2 with iff (6fb7f.. x0) (6fb7f.. x1), x0 = x1 leaving 2 subgoals.
Apply H0 with iff (6fb7f.. x0) (6fb7f.. x1)x0 = x1 leaving 2 subgoals.
Assume H2: x0 = ChurchBoolTru.
Apply H2 with λ x2 x3 : ι → ι → ι . iff (6fb7f.. x3) (6fb7f.. x1)x3 = x1.
Apply H1 with iff (6fb7f.. ChurchBoolTru) (6fb7f.. x1)ChurchBoolTru = x1 leaving 2 subgoals.
Assume H3: x1 = ChurchBoolTru.
Apply H3 with λ x2 x3 : ι → ι → ι . iff (6fb7f.. ChurchBoolTru) (6fb7f.. x3)ChurchBoolTru = x3.
Let x2 of type (ιιι) → (ιιι) → ο be given.
The subproof is completed by applying H5.
Assume H3: x1 = ChurchBoolFal.
Apply H3 with λ x2 x3 : ι → ι → ι . iff (6fb7f.. ChurchBoolTru) (6fb7f.. x3)ChurchBoolTru = x3.
Apply FalseE with ChurchBoolTru = ChurchBoolFal.
Apply H4 with False.
Apply unknownprop_4db3dccc9d2b781cbc51e143c21b1ce8ea7a94ab506258592ed1c524bf6deaea.
Apply H6.
The subproof is completed by applying unknownprop_c16a7e66d013fefc3e2d9d08fe341ba71aa55df92f5de99da11396ce50578700.
Assume H2: x0 = ChurchBoolFal.
Apply H2 with λ x2 x3 : ι → ι → ι . iff (6fb7f.. x3) (6fb7f.. x1)x3 = x1.
Apply H1 with iff (6fb7f.. ChurchBoolFal) (6fb7f.. x1)ChurchBoolFal = x1 leaving 2 subgoals.
Assume H3: x1 = ChurchBoolTru.
Apply H3 with λ x2 x3 : ι → ι → ι . iff (6fb7f.. ChurchBoolFal) (6fb7f.. x3)ChurchBoolFal = x3.
Apply FalseE with ChurchBoolFal = ChurchBoolTru.
Apply H4 with False.
Apply unknownprop_4db3dccc9d2b781cbc51e143c21b1ce8ea7a94ab506258592ed1c524bf6deaea.
Apply H5.
The subproof is completed by applying unknownprop_c16a7e66d013fefc3e2d9d08fe341ba71aa55df92f5de99da11396ce50578700.
Assume H3: x1 = ChurchBoolFal.
Apply H3 with λ x2 x3 : ι → ι → ι . iff (6fb7f.. ChurchBoolFal) (6fb7f.. x3)ChurchBoolFal = x3.
Let x2 of type (ιιι) → (ιιι) → ο be given.
The subproof is completed by applying H5.
Assume H2: x0 = x1.
Apply H2 with λ x2 x3 : ι → ι → ι . iff (6fb7f.. x3) (6fb7f.. x1).
The subproof is completed by applying iff_refl with 6fb7f.. x1.