Let x0 of type ο be given.
Let x1 of type ο be given.
Assume H0: x0 = x1 ⟶ ∀ x2 : ο . x2.
Let x2 of type ο be given.
Apply xm with
x2,
or (x2 = x0) (x2 = x1) leaving 2 subgoals.
Assume H1: x2.
Apply xm with
x0,
or (x2 = x0) (x2 = x1) leaving 2 subgoals.
Assume H2: x0.
Apply orIL with
x2 = x0,
x2 = x1.
Apply prop_ext_2 with
x2,
x0 leaving 2 subgoals.
Assume H3: x2.
The subproof is completed by applying H2.
Assume H3: x0.
The subproof is completed by applying H1.
Apply xm with
x1,
or (x2 = x0) (x2 = x1) leaving 2 subgoals.
Assume H3: x1.
Apply orIR with
x2 = x0,
x2 = x1.
Apply prop_ext_2 with
x2,
x1 leaving 2 subgoals.
Assume H4: x2.
The subproof is completed by applying H3.
Assume H4: x1.
The subproof is completed by applying H1.
Apply FalseE with
or (x2 = x0) (x2 = x1).
Apply H0.
Apply prop_ext_2 with
x0,
x1 leaving 2 subgoals.
Assume H4: x0.
Apply FalseE with
x1.
Apply H2.
The subproof is completed by applying H4.
Assume H4: x1.
Apply FalseE with
x0.
Apply H3.
The subproof is completed by applying H4.
Apply xm with
x0,
or (x2 = x0) (x2 = x1) leaving 2 subgoals.
Assume H2: x0.
Apply xm with
x1,
or (x2 = x0) (x2 = x1) leaving 2 subgoals.
Assume H3: x1.
Apply FalseE with
or (x2 = x0) (x2 = x1).
Apply H0.
Apply prop_ext_2 with
x0,
x1 leaving 2 subgoals.
Assume H4: x0.
The subproof is completed by applying H3.
Assume H4: x1.
The subproof is completed by applying H2.
Apply orIR with
x2 = x0,
x2 = x1.
Apply prop_ext_2 with
x2,
x1 leaving 2 subgoals.
Assume H4: x2.
Apply FalseE with
x1.
Apply H1.
The subproof is completed by applying H4.
Assume H4: x1.
Apply FalseE with
x2.
Apply H3.
The subproof is completed by applying H4.
Apply orIL with
x2 = x0,
x2 = x1.
Apply prop_ext_2 with
x2,
x0 leaving 2 subgoals.
Assume H3: x2.
Apply FalseE with
x0.
Apply H1.
The subproof is completed by applying H3.
Assume H3: x0.
Apply FalseE with
x2.
Apply H2.
The subproof is completed by applying H3.