Is it safe to install a three pronged cable into a four pronged dryer?
What you are doing is done all the time, and there is no problem with it, if done properly.   The fact is these 3 prong receptacles still exists in many older homes, and there is no requirement to upgrade an entire circuit simply to plug in a device. If you go out and purchase a new electric dryer, the seller will ask if you have a 3 or 4 prong receptacle. They will then sell you the appropriate cord, based on your answer (if you don't know, they'll probably sell you both and tell you to return the one you don't need, or offer their instillation services).
RTFM
If you check the manufacturers installation instructions, there should be a procedure for connecting the dryer to a 3 prong receptacle. The procedure can very from manufacturer to manufacturer, so make sure you check the documentation for your specific dryer.
From a random whirlpool manual:

This dryer is manufactured ready to install with a 3-wire  electrical
supply connection. The neutral ground conductor  is permanently
connected to the neutral conductor (white  wire) within the dryer. If
the dryer is installed with a 4-wire  electrical supply connection,
the neutral ground conductor  must be removed from the external ground
connector (green  screw), and secured under the neutral terminal
(center or  white wire) of the terminal block. When the neutral ground
conductor is secured under the neutral terminal (center or  white
wire) of the terminal block, the dryer cabinet is isolated from the
neutral conductor.

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