Let x0 of type ι be given.
Let x1 of type ι → ι be given.
Let x2 of type ι → ι → ο be given.
Let x3 of type ι → ι → ι be given.
Let x4 of type ι be given.
Apply ReplSep2E_impred with
x0,
x1,
x2,
x3,
x4,
∃ x5 . and (x5 ∈ x0) (∃ x6 . and (x6 ∈ x1 x5) (and (x2 x5 x6) (x4 = x3 x5 x6))) leaving 2 subgoals.
The subproof is completed by applying H0.
Let x5 of type ι be given.
Assume H1: x5 ∈ x0.
Let x6 of type ι be given.
Assume H2: x6 ∈ x1 x5.
Assume H3: x2 x5 x6.
Assume H4: x4 = x3 x5 x6.
Let x7 of type ο be given.
Assume H5:
∀ x8 . and (x8 ∈ x0) (∃ x9 . and (x9 ∈ x1 x8) (and (x2 x8 x9) (x4 = x3 x8 x9))) ⟶ x7.
Apply H5 with
x5.
Apply andI with
x5 ∈ x0,
∃ x8 . and (x8 ∈ x1 x5) (and (x2 x5 x8) (x4 = x3 x5 x8)) leaving 2 subgoals.
The subproof is completed by applying H1.
Let x8 of type ο be given.
Assume H6:
∀ x9 . and (x9 ∈ x1 x5) (and (x2 x5 x9) (x4 = x3 x5 x9)) ⟶ x8.
Apply H6 with
x6.
Apply andI with
x6 ∈ x1 x5,
and (x2 x5 x6) (x4 = x3 x5 x6) leaving 2 subgoals.
The subproof is completed by applying H2.
Apply andI with
x2 x5 x6,
x4 = x3 x5 x6 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.