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If we are trusting in the gift God gave us, then yes. He did all the work.
He did the giving of the ability to know the gospel is true.
We do the believing in response to him showing us the gospel is true. He does not do our believing for us. He gives the faith so we can do our believing, but he does not do our believing.
We can choose to stop believing. The Galatians and the Hebrew church, and perhaps the Corinthians too, are proof of this. Paul had to go into damage control mode and bring them back to belief/trust in Christ.