What does it mean to bear witness to Tawheed?
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Sheikh Muhammed Salih Al-Munajjid saied:
What it means to bear witness that there is no god except Allaah is to deny that anyone other than Allaah has the right to be worshipped, and to affirm that this is the right of Allaah alone, with no partner or associate.
Allaah says (interpretation of the meaning):
“That is because Allaah He is the Truth (the only True God of all that exists, Who has no partners or rivals with Him), and what they (the polytheists) invoke besides Him, it is Baatil (falsehood). And verily, Allaah He is the Most High, the Most Great”[al-Hajj 22:62]
So the words laa ilaaha (there is no god) deny all things that are worshipped instead of Allaah, and the words illa Allaah (except Allaah) affirm that all kinds of worship are for Allaah alone. So what they mean is that there is none that is rightfully worshipped except Allaah.
Just as Allaah has no partner in His dominion, so too He has no partner in His worship.
What it means to bear witness that Muhammad
A person can only enter Islam by uttering this twin testimony of faith; its two parts are intertwined and cannot be separated. For the conditions of testifying that there is no god but Allaah are the same as the conditions of testifying that Muhammad is the Messenger of Allaah.
These, along with the evidence for them, are mentioned in these theards
Conditions of the acceptance of the Shahaadatayn
The Conditions of Laa ilaaha illa-Allaah