It depends on how visible your child navigation is. While Breadcrumbs can be redundant in some cases, they do serve as a medium of quick navigation as well. 

To quote this article [Breadcrumb Navigation Examined: Best Practices & Examples][1]


> Fact of the matter is, breadcrumb navigation will greatly increase the
> usability of a website. Breadcrumbs give users an alternative method
> of navigation, allow them to see where they stand in the hierarchy of
> a website, and will reduce the number of steps needed to navigate to a
> higher-level within a website.

This article from smashing Magazine [Breadcrumbs In Web Design: Examples And Best Practices][2] has this to say : 

> Here are just some of the benefits of using a breadcrumb trail.
> 
> **Convenient for users** Breadcrumbs are used primarily to give users a secondary means of navigating a website. By offering a breadcrumb
> trail for all pages on a large multi-level website, users can navigate
> to higher-level categories more easily.
> 
> **Reduces clicks or actions to return to higher-level pages**
> 
> Instead of using the browser’s “Back” button or the website’s primary
> navigation to return to a higher-level page, users can now use the
> breadcrumbs with a fewer number of clicks.
> 
> **Doesn’t usually hog screen space**.
> 
> Because they’re typically horizontally oriented and plainly styled,
> breadcrumb trails don’t take up a lot of space on the page. The
> benefit is that they have little to no negative impact in terms of
> content overload, and they outweigh any negatives if used properly.
> 
> **Reduces bounce rates** Breadcrumb trails can be a great way to entice first-time visitors to peruse a website after having viewed the
> landing page. For example, say a user arrives on a page through a
> Google search, seeing a breadcrumb trail may tempt that user to click
> to higher-level pages to view related topics of interests. This, in
> turn, reduces the overall website bounce rate.


  [1]: http://www.hongkiat.com/blog/breadcrumb-navigation-examined-best-practices-examples/
  [2]: http://www.smashingmagazine.com/2009/03/17/breadcrumbs-in-web-design-examples-and-best-practices-2/