Second: the most that can be inferred is that covering is a condition for the validity of a woman’s prayer alone, which is narrower than the general claim. As for extending the ruling to men, the analogy fails, because the provocation caused by a woman’s exposure is a consideration that does not exist with a man’s ʿawrah (private parts).
Third: the ḥadīth of Sahl ibn Saʿd, related by the Two Shaykhs, Abū Dāwūd, and al-Nasāʾī, states: “The men used to pray with the Prophet ﷺ with their waist-wraps tied around their shoulders like boys, and it would be said to the women, ‘Do not raise your heads until the men have sat up.’” Abū Dāwūd adds: “because the waist-wraps were tight.” This indicates that covering was not obligatory, much less a condition for validity.
Fourth: the ḥadīth of ʿAmr ibn Salamah, in which he says, “I used to lead them in prayer wearing a split mantle; when I prostrated it would draw up from me”—in another wording: “my backside became exposed.” A woman of the clan said, “Will you not shield us from the backside of your reciter?” … The report is found in al-Bukhārī, Abū Dāwūd, and al-Nasāʾī.
The truth, then, is that covering the ʿawrah in prayer is merely an obligation like any other circumstance, not a condition whose absence would render the prayer invalid. Those who deny conditionality advance flimsy juristic arguments, such as their claim: “If covering were a condition of prayer it would apply only to prayer, and it would require an intention.” The first premise is overturned by faith (īmān), which is a condition for prayer yet not confined to it; the second by facing the qiblah, which requires no intention; and the third by the case of one who is unable to recite or glorify, for he prays in silence.” End of his words—may Allah have mercy on him.