bleach and bleach substitute

I still figure that bleach substitute is not nearly as good as bleach when white clothes are involved, and I should use bleach with really dirty white clothes, or stuff that didn't come clean with a regular wash.
But is it ok to add bleach when using a detergent that already includes bleach-substitute (I can't remember what they call that)?
Should I have a box of non-bleach detergent around for using with bleach?
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wrote:

No. And products such as OxiClean are not as effective as (inexpensive) household bleach in the laundry.

Yes. Be extra careful with household bleach, a little goes a long way and too much can damage fabrics.

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