Proof involving invertible matrices

I need to prove that if A is an invertible matrix and B is row equivalent to A, then B is also invertible.

I understand the logic of that statement, but I'm not experienced enough to know how to write it into words that prove it.

Here's my thinking so far:

1.) B is row equivalent to A, therefore there exist a sequence of elementary row operations that can be performed on A to produce B:

(En*...*E2*E1)A=B

2.) A is invertible, therefore there exist a sequence of elementary row operations that can be performed on A to produce the identity matrix:

(En*...*E2*E1)A=I

3.) From 1 & 2, there exist a sequence of elementary row operations that can be performed on B to produce the identity matrix, therefore B is also invertible.

How can I write the above statements so that they flow in a step-by-step proof?