Let x0 of type ι be given.
Let x1 of type ι → ο be given.
Let x2 of type ι → ο be given.
Assume H0:
∀ x3 . x3 ∈ x0 ⟶ iff (x1 x3) (x2 x3).
Apply set_ext with
Sep x0 x1,
Sep x0 x2 leaving 2 subgoals.
Let x3 of type ι be given.
Assume H1:
x3 ∈ Sep x0 x1.
Apply SepE with
x0,
x1,
x3,
x3 ∈ Sep x0 x2 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2: x3 ∈ x0.
Assume H3: x1 x3.
Apply SepI with
x0,
x2,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply H0 with
x3,
x2 x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H4: x1 x3 ⟶ x2 x3.
Assume H5: x2 x3 ⟶ x1 x3.
Apply H4.
The subproof is completed by applying H3.
Let x3 of type ι be given.
Assume H1:
x3 ∈ Sep x0 x2.
Apply SepE with
x0,
x2,
x3,
x3 ∈ Sep x0 x1 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2: x3 ∈ x0.
Assume H3: x2 x3.
Apply SepI with
x0,
x1,
x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Apply H0 with
x3,
x1 x3 leaving 2 subgoals.
The subproof is completed by applying H2.
Assume H4: x1 x3 ⟶ x2 x3.
Assume H5: x2 x3 ⟶ x1 x3.
Apply H5.
The subproof is completed by applying H3.