∃x ∀y R(x, y)                                 [Hypothesis]

Let c be a constant such that ∀y R(c, y)
    ∀y R(c, y)

    Given constant d:
        R(c, d)                               [∀ elim]
        ∃x R(x, d)                            [∃ intro]

    ∀y ∃x R(x, y)                             [∀ intro]

∀y ∃x R(x, y)                                 [∃ elim]