Let x0 of type ο be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Apply xm with
x0,
or (and x0 (If_i x0 x1 x2 = x1)) (and (not x0) (If_i x0 x1 x2 = x2)) leaving 2 subgoals.
Assume H0: x0.
Claim L1:
or (and x0 (x1 = x1)) (and (not x0) (x1 = x2))Apply orIL with
and x0 (x1 = x1),
and (not x0) (x1 = x2).
Apply andI with
x0,
x1 = x1 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι → ι → ο be given.
Assume H1: x3 x1 x1.
The subproof is completed by applying H1.
Apply Eps_i_ax with
λ x3 . or (and x0 (x3 = x1)) (and (not x0) (x3 = x2)),
x1.
The subproof is completed by applying L1.
Claim L1:
or (and x0 (x2 = x1)) (and (not x0) (x2 = x2))Apply orIR with
and x0 (x2 = x1),
and (not x0) (x2 = x2).
Apply andI with
not x0,
x2 = x2 leaving 2 subgoals.
The subproof is completed by applying H0.
Let x3 of type ι → ι → ο be given.
Assume H1: x3 x2 x2.
The subproof is completed by applying H1.
Apply Eps_i_ax with
λ x3 . or (and x0 (x3 = x1)) (and (not x0) (x3 = x2)),
x2.
The subproof is completed by applying L1.