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Is there a way to differentiate between 'which implies' and 'implies' in mathematical notation?

I write $\implies$ for both which sometimes feels wrong and lazy to me especially when dealing with logic.

For example:

'$0 = 1$ which implies $0 = 0$' is different from saying '$0 = 1$ implies $0 = 0$'. The former is false (I assert the first statement '$0 = 1$' as well as the implication) while the latter is true.

On paper I would just write for both $$0 = 1 \implies 0 = 0$$

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    The usage you describe could just be replaced with the word "and." Note that $P\wedge (P\implies Q) \equiv P\wedge (\neg P\vee Q) \equiv (P\wedge \neg P)\vee (P\wedge Q) \equiv P\wedge Q$. In practice, though, I often see the phrases used interchangeably without the additional implication that the hypothesis must be true, in particular if being used in the middle of a string of implications. – 2017-02-21
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    If you must have a symbol for "which implies" , how about $\therefore$? – 2017-02-21

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