Your were just near to your result and looking into your code looks like you are learning, Good Work!! you have to check just if any one of the content is enable so you need to hide All of the three buttons and display your Sub contents. Below is the correct code as per your requirement,
<!-- Display all of the SubContents is disable. -->
<div *ngIf="!showSubContent && !showSubContentTwo && !showSubContentThree">
<button (click)="showSubContent=!showSubContent">Show Sub content1</button>
<button (click)="showSubContentTwo=!showSubContentTwo">Show Sub content2</button>
<button (click)="showSubContentThree=!showSubContentThree">Show Sub content3</button>
<h2> Main content </h2>
</div>
<!-- Display if SubContent-1 is enable. -->
<div *ngIf="showSubContent && !showSubContentTwo && !showSubContentThree">
Sub Content 1 here
<button (click)="showSubContent=!showSubContent">Show Main Content</button>
</div>
<!-- Display if SubContent-2 is enable. -->
<div *ngIf="showSubContentTwo && !showSubContent && !showSubContentThree">
Sub Content 2 here
<button (click)="showSubContentTwo=!showSubContentTwo">Show Main Content</button>
</div>
<!-- Display if SubContent-3 is enable. -->
<div *ngIf="showSubContentThree && !showSubContentTwo && !showSubContent">
Sub Content 3 here
<button (click)="showSubContentThree=!showSubContentThree">Show Main Content</button>
</div>