Although this will be a bit slower, you can also use the array formula (by pressing Ctrl + Shift + Enter after entering it into the cell):
{=IF(SUM(COUNTIF(A1:C1,A1:C1))=POWER(COUNTA(A1:C1), 2),"true","false")}
This will work even if there are empty cells in the range.
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UPDATE:
This array formula also works, making sure that there is only one unique value in the range. It also still returns the exact value, even if there are spaces in the space:
{=SUM(IFERROR(1 / COUNTIF(A2:C2,A2:C2), 0)) = 1}
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