Neither of wallets keep your coins. They keep the pair of private - public keys. All you need to get bcc and btc after 1 st Aug is to import your private key into relevant wallet. One of them will be solely for btc the other one for bcc
I got some PM asking to give more details on this. So, actually u are keeping btc not in the wallet but on the address. Keep your btc on any address u control with the private key. As an option u may go to https://www.bitaddress.org/ generate a pair of private-public (the last of the pair is your address) key (write it down and keep in the safe place) and send your btc to the address (public key) u generated. Then, when needed, you may import the private key in any wallet u choose. After Aug 1st when the all settle down you may import the private key relevant to your address to the wallet that synchronises with btc chain (to get control of your btc) and at same time to another wallet that synchronises with bcc chain (to get control of your bcc)
Thanks for that info. I have a bitaddress generated paper wallet with private and public keys. I have just withdrawn my BTC from the exchange to the paper wallet and it's pending confirmations. So, if I am understanding this correctly, you can spend your BTC on both the chains simultaneously after the split, right? Kind of like double spending, one on each chain?