Question

Which scripture is the source of the verse “janmana jayate shudrah.. " (everyone is born a shudra)?

Answer

No this shloka is not from Shreemad Bhagvat Purana.

This shloka is given in Skanda Purana Vol.18 Book VI , Nagar Kanda , Chapter 239 , ( Efficacy of Adoration , Penance etc.) Verse 31-34

जन्मना जायते शूद्रः संस्कारात् द्विज उच्यते।
शापानुग्रहसामर्थ्यं तथा क्रोधः प्रसन्नता ।|31-34||

31 : A Man is no better than a sudra at his birth .He is called Brahmana (Twice Born) due to the consecration. The ability to curse and to bless, the state of being angry and pleased and the status of being the foremost in all three worlds occur only in Brahmana

This chapter is about adoration of Lord Vishnu is performed by means of Sixteen Upacharas. (Shodasopachara puja) (षोडशोपचार पूजा). This shloka is used in context of Achamana ,Yajnopavita , Dvija's. But the specific chapter talks about Devotion , Chatrumas puja and Panchyatana puja.

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