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An ideal controlled-NOT gate followed by projective measurements can be used to identify specific Bell states of its two input qubits. When the input qubits are each members of independent Bell states, these projective measurements can be used to swap the post-selected entanglement onto the remaining two qubits. Here we apply this strategy to produce heralded two-photon polarization entanglement using Bell states that originate from independent parametric down-conversion sources, and a particular probabilistic controlled-NOT gate that is constructed from linear optical elements. The resulting implementation is closely related to an earlier proposal by Sliwa and Banaszek, and can be intuitively understood in terms of familiar quantum information protocols. The possibility of producing a "pseudo-demand" source of two-photon entanglement by storing and releasing these heralded pairs from independent cyclical quantum memory devices is also discussed.