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Your intuition is correct. Let's suppose that at STP, a human needs to breathe 1L of O2 in 200 minutes. That's 1.43g of gaseous oxygen. If the pressure is raised to 5 bar, a human still needs 1.43g of O2 -- but it will only take up 0.2L of volume, and the other neutral gas mass will have to increase as well.

Similarly, a fire that consumes 143g of O2 and 90g of powdered aluminum will use those proportions regardless of whether the O2 is mixed in a higher or lower pressure of neutral gas -- the gating factor is how fast the oxygen can come in contact with the aluminum.