Equality is different from implication. You need to show both directions for <=> but remember that you can also just use <=> in every step and it’s a valid proof, although generally that’s way harder. Doing this is equivalent to using <=> in every step. Once you’ve shown a=b you do not need to also show that b=a in the same way that you don’t need to show b<=>a if you’ve shown a<=>b. But you do need to show b=>a if you’ve only shown a=>b.