"by picking 4 different index funds to diversify"
Some diversification is "don't be left behind", like holding both a US index fund and an international index fund. Other diversification aims at cushioning against stock market corrections, like a US bond fund.
Something else to think about: some indexes can be bought as one fund, or several. For example, Vanguard Total Stock Market captures the entire US stock market. But you can also buy 3 funds separately: a large cap, mid cap and small cap index fund. You wind up with the same assets in both cases, although with 3 funds you might not hold the same weightings as the market.
An all in one fund should perform no differently than holding it's underlying funds, except for the management fee difference. It might help if you list the 4 funds you are considering, so it becomes a less abstract discussion.