The phrase, ‘Alhamdulillah is an all inclusive phrase, it is not merely a declaration of thanks---since thanking does not require nor imply praise---rather it contains a greater meaning. Ibrahim alayhi as-salaam thanked his father because Allah commanded him to do so, but he did not praise him. Allah is worthy of being praised and thanked at the same time. Therefore, in the phrase, ‘Alhamdulillah,’ praise and thanks are combined.
Hence, an accurate translation of that part of the ayah would be, “All praise and thanks be to Allah.” Note, that the verbal-noun ‘praise’ is mentioned here as a noun and not a verb (e.g.praise Allah). This subtle feature implies permanence, as nouns imply permanence and verb simply time-constraints and temporariness. And this is appropriate because the praise of Allah is timeless, it was there before we were here, it is here now, and will remain here after us. So when we say, ‘Alhamdulillah,’ we are declaring that all praise and thanks are for Allah, and they are independent of us.