Encryption technologies have become one of the most prevalent solutions to safeguard data confidentiality in many real-world applications, e.g., cloud-based data storage systems. Encryption outputting a relatively “static” format of encrypted data, however, may hinder further data operations. For example, encrypted data may need to be “transformed” into other formats for computation or other purposes. To enable encryption to be used in another device equipped with a different encryption mechanism, the concept of encryption switching was first proposed in CRYPTO 2016 for conversion particularly between Paillier and ElGamal encryptions. This paper considers the conversion between conventional identity-based and attribute-based encryptions and further proposes a concrete construction via the technique of proxy re-encryption. The construction is proved to be CPA secure in the standard model under q-decisional parallel bilinear Diffie-Hellman exponent assumption. The performance comparisons highlight that our bridging mechanism reduces computation and communication cost on the client side, especially when the data of the client is encrypted and outsourced to a remote cloud. The computational costs with respect to re-encryption (on the server side) and decryption (on the client side) are acceptable in practice.