Can you use phyrexian mana to pay for a spell without using any colored mana?

Phyrexian mana symbols (written as {W/P}, {U/P}, {R/P}, etc.) each offer two payment options: pay one mana of the appropriate color, OR pay 2 life. You are never forced to pay the colored mana — the life payment is always available as a full substitute. (CR 107.4f)

This means a card with only Phyrexian mana in its cost, such as Gut Shot ({R/P}), can be cast for free in terms of mana — you simply pay 2 life. No mana of any color is required at all.

If a spell has both Phyrexian mana symbols and regular mana symbols (e.g., {1}{U/P}{U/P}), you can pay 2 life for each Phyrexian symbol, but you must still pay the regular {1} with mana. Each symbol is paid independently. (CR 107.4f, 601.2f)

Concrete example: You cast Gitaxian Probe ({U/P}) with no blue mana available. You choose to pay 2 life instead of {U}. The spell is cast without spending any mana whatsoever — entirely legal.

Note that paying life for Phyrexian mana is still subject to normal restrictions: you cannot pay life this way if an effect prevents you from paying life as a cost, and you must have enough life (though you can pay down to 1 life; you cannot pay life you don't have). (CR 119.4)