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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ιο be given.
Let x2 of type ι be given.
Assume H0: prim1 x2 x0.
Apply iffI with x1 x2, (λ x3 . and (x1 x3) (x3 = x0∀ x4 : ο . x4)) x2 leaving 2 subgoals.
Assume H1: x1 x2.
Apply andI with x1 x2, x2 = x0∀ x3 : ο . x3 leaving 2 subgoals.
The subproof is completed by applying H1.
Assume H2: x2 = x0.
Apply In_irref with x0.
Apply H2 with λ x3 x4 . prim1 x3 x0.
The subproof is completed by applying H0.
Assume H1: (λ x3 . and (x1 x3) (x3 = x0∀ x4 : ο . x4)) x2.
Apply H1 with x1 x2.
Assume H2: x1 x2.
Assume H3: x2 = x0∀ x3 : ο . x3.
The subproof is completed by applying H2.