Yes, there are studies done on it and proven.
A new study finds that kids and teens who are more religious or spiritual do better as they age.
www.forbes.com
Did you read the actual study? It's certainly relatively well-produced and in-depth research, but I doubt the authors would claim that it's "proof" of anything. If you look into it, while it's a relatively widespread (but largely white, American, middle class) study, it's entirely based on self-reporting. So it's not a surprise that someone who claims to go to church every week on a survey will also self report lower numbers of sexual partners, lower use of illegal drugs, lower consumption of alcohol, and higher levels of things like volunteering. One of the survey questions on forgiveness is as follows: "Because of my spiritual or religious beliefs, I have forgiven those who hurt me." How is that not clearly a loaded question? Someone who isn't religious who has forgiven those who have wronged them still couldn't answer yes to that question. When self-reporting, even when anonymous, people will answer what they think they should answer. There have been surveys that show that more people "go to church every week" in America than there are churches to fit them all in.
Some of the medical problems are likely to be more reliable because there's not the same religious stigma about things like cancer or obesity, but self-reporting things like "feeling depressed" isn't exactly reliable, and even diagnosis of depression could be more of an indicator of how likely someone is to seek medical help than actual levels of depression. People in a particularly close-knit church community might be more likely to go to a pastor than a doctor if they're feeling depressed, for example. Some of the questions are well-written to try and remove any religious bias, but some aren't especially successful at doing so.
Some of the indicators on the study are literally lifted straight from religious beliefs. So for example, there's absolutely no reason that "number of sexual partners" should be framed as a negative thing, and yet there it in the list of health problems along with depression, prescription drug addiction and cancer. What you're saying there is that people who self-report religious beliefs will also self-report following those religious beliefs. You might as well put "not covering your head" as a problem that needs to be surveyed and then ask a bunch of Muslim women how often they do it. Basically if you ask a person how often they attend religious services and pray, and then on the same sheet of paper ask them how often they smoke marijuana and how many people they've fucked, you're not necessarily going to get a truthful answer every time.
Which isn't to dismiss the study, just show that it's a single study of a single demographic in a single country, and therefore isn't proof of anything.