I have a I phone 4 that is jailbroken at 4.3.1. can I update it and still jailbreak it on5.1.1or 6.

I have a I phone 4 that is jailbroken at 4.3.1. can I update it and still jailbreak it on5.1.1or 6.0.1 un tethered. Now does anyone know if it is still possible to update it to 5.1.1 even though it is no longer signed by apple. Thanks in advance

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