Is my ghusl in the bathtub valid?

Hello so there have been a lot of thunderstorms recently which is why I have been kind of scared to do ghusl in the shower. Because of that, I took a cup of water to rinse my nose & mouth. Then I filled a bathtub with water and washed my whole body with it once by immersing myself in it and pouring that same water over myself with a little bucket several times so that it reaches all parts of my body. I didn’t do wudu afterwards, I just got up and prayed. Was my ghusl valid? I really don’t want to take showers during thunderstorms so I didn’t know what else to do.