Mirror formatting

I am trying to mirror cells (both content and formatting) across sheets.
For example, on Sheet 2, I enter “=$Sheet1.A1” and it mirrors the contents. I can copy and paste-special the formatting, but then if I later change the color of the Sheet 1.A1, the recipient on Sheet 2 doesn’t change.
Conditional formatting doesn’t work in my case as the colors I choose are not related to the content or value.
How can I accomplish this?

You could create a cell style; making a change in that cell style will update all cells of that style.
In attached spreadsheet, in the Sidebar, Styles, right click the cell style MyStyle and change the background to blue
ChangeCellStyleToChangeAll.ods (17.9 KB)

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Thank you for your suggestion. In this case it won’t work simply because that will change everything of that style.
I have over 100 rows. Column A has quantity as the cell data and quality is denoted as either red lettering on black background (we’ll call this “Style 1” for our purposes) or has black lettering on red background (we’ll call this style 2). Column B has either yellow lettering on black background or black lettering on yellow background (styles 3 and 4). Column C has green on black or black on green (styles 5 & 6)
I have several sheets that I want to match so no matter which sheet I’m working with I know exactly the quantity and quality I have of each. Let’s say the quality of cell 4A changes from style 1 to style 2. I can’t just make the change in the overall style or it will change everything in column.
What I want is to designate on sheet 2, =$Sheet1.A1(Style 1,ThenStyle 1,Else Style 2) all while maintaining the =$Sheet1.A1 which mirrors the value. I probably have the above syntax wrong but that’s what I hoping to get clarification…