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If this plan is carried out smoothly, I'm convinced that...

Once this plan is carried out smoothly, I'm convinced that...

What's the difference between those two sentences in meaning? I think it almost sounds the same.

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If the plan is carried out smoothly.....

The conjunction "if" is also used in imaginary and conditional sentences. It's used for saying that one thing can or will happen depending on the happening of something else. So this sentence is conditional.

Once the plan is carried out smoothly.....

This sentence is quite different from the former sentence. It's not conditional. The conjunction "once" in this sentence means "as soon as or when".

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If this plan is carried out smoothly, I'm convinced that...

The plan may not be carried out smoothly. If it is, the "I'm convinced that..." part will be true.

Once this plan is carried out smoothly, I'm convinced that...

You are waiting until the plan is carried out smoothly. At some point eventually the "I'm convinced that..." is expected to be true.

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if talks about the 'condition'. once talks about 'the first step'.

Thus,

If this plan is carried out smoothly, I'm convinced that...

is a condition of you getting convinced only if the plan is carried out smoothly.

And...

Once this plan is carried out smoothly, I'm convinced that...

talks about the first step of the plan. And, success of this will convince you.

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