"If I Were A Rich Man"

Is this correct grammar?

Shouldn't Tevye be saying "If I was a rich man?"

Actually, no!

"I was" is correct when referring to a fact or a real situation: "If I was awake I would have watched the game"

But when talking about an impossible situation, such as Tevye being a rich man, it is correct to use "if I were."  The use of the plural form "were" instead of the usually correct singular "was" is like a signpost telling the reader (or listener) that the situation described in the sentence is unrealistic.

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