Let x0 of type ο be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Let x3 of type ο be given.
Assume H0:
x0 ⟶ If_i x0 x1 x2 = x1 ⟶ x3.
Assume H1:
not x0 ⟶ If_i x0 x1 x2 = x2 ⟶ x3.
Apply unknownprop_eb8e8f72a91f1b934993d4cb19c84c8270f73a3626f3022b683d960a7fef89cb with
and x0 (If_i x0 x1 x2 = x1),
and (not x0) (If_i x0 x1 x2 = x2),
x3 leaving 3 subgoals.
The subproof is completed by applying unknownprop_dd02c0c510e8f3ca2e39c3596405da8ac0db80e9b43461e8e498428b58b79272 with x0, x1, x2.
Assume H2:
and x0 (If_i x0 x1 x2 = x1).
Apply andE with
x0,
If_i x0 x1 x2 = x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H0.
Apply andE with
not x0,
If_i x0 x1 x2 = x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H1.