Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type ι be given.
Assume H0:
x2 ∈ UPair x0 x1.
Apply UPairE with
x2,
x0,
x1,
x2 ∈ {If_i (x0 ∈ x3) x0 x1|x3 ∈ prim4 (prim4 x0)} leaving 3 subgoals.
The subproof is completed by applying H0.
Assume H2: x2 = x0.
Apply H2 with
λ x3 x4 . x4 ∈ {If_i (x0 ∈ x5) x0 x1|x5 ∈ prim4 (prim4 x0)}.
Apply If_i_1 with
x0 ∈ prim4 x0,
x0,
x1,
λ x3 x4 . x3 ∈ {If_i (x0 ∈ x5) x0 x1|x5 ∈ prim4 (prim4 x0)} leaving 2 subgoals.
The subproof is completed by applying Self_In_Power with x0.
Apply ReplI with
prim4 (prim4 x0),
λ x3 . If_i (x0 ∈ x3) x0 x1,
prim4 x0.
The subproof is completed by applying Self_In_Power with
prim4 x0.
Assume H2: x2 = x1.
Apply H2 with
λ x3 x4 . x4 ∈ {If_i (x0 ∈ x5) x0 x1|x5 ∈ prim4 (prim4 x0)}.
Apply If_i_0 with
x0 ∈ 0,
x0,
x1,
λ x3 x4 . x3 ∈ {If_i (x0 ∈ x5) x0 x1|x5 ∈ prim4 (prim4 x0)} leaving 2 subgoals.
The subproof is completed by applying EmptyE with x0.
Apply ReplI with
prim4 (prim4 x0),
λ x3 . If_i (x0 ∈ x3) x0 x1,
0.
The subproof is completed by applying Empty_In_Power with
prim4 x0.