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 prop_ext_2 with
∃ x3 . and (x3 ∈ x0) (x2 x3),
∃ x3 . and (x3 ∈ x0) (x1 x3) leaving 2 subgoals.
Assume H1:
∃ x3 . and (x3 ∈ x0) (x2 x3).
Apply H1 with
∃ x3 . and (x3 ∈ x0) (x1 x3).
Let x3 of type ι be given.
Assume H2:
(λ x4 . and (x4 ∈ x0) (x2 x4)) x3.
Apply H2 with
∃ x4 . and (x4 ∈ x0) (x1 x4).
Assume H3: x3 ∈ x0.
Assume H4: x2 x3.
Let x4 of type ο be given.
Assume H5:
∀ x5 . and (x5 ∈ x0) (x1 x5) ⟶ x4.
Apply H5 with
x3.
Apply andI with
x3 ∈ x0,
x1 x3 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x3,
x1 x3 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H6: x1 x3 ⟶ x2 x3.
Assume H7: x2 x3 ⟶ x1 x3.
Apply H7.
The subproof is completed by applying H4.
Assume H1:
∃ x3 . and (x3 ∈ x0) (x1 x3).
Apply H1 with
∃ x3 . and (x3 ∈ x0) (x2 x3).
Let x3 of type ι be given.
Assume H2:
(λ x4 . and (x4 ∈ x0) (x1 x4)) x3.
Apply H2 with
∃ x4 . and (x4 ∈ x0) (x2 x4).
Assume H3: x3 ∈ x0.
Assume H4: x1 x3.
Let x4 of type ο be given.
Assume H5:
∀ x5 . and (x5 ∈ x0) (x2 x5) ⟶ x4.
Apply H5 with
x3.
Apply andI with
x3 ∈ x0,
x2 x3 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x3,
x2 x3 leaving 2 subgoals.
The subproof is completed by applying H3.
Assume H6: x1 x3 ⟶ x2 x3.
Assume H7: x2 x3 ⟶ x1 x3.
Apply H6.
The subproof is completed by applying H4.