I generally agree with Plato and topsquark here, but I'll give this a shot.
I.3 does not say that. You rearranged the sentence to make it say that. The original I.3 isOriginally Posted by OReilly
Take an example. Suppose line L contains distinct points A and B and line L' contains different distinct points A' and B'. Then L and L' are both lines that contain two distinct points. So by I.3, L = L'. But that is not the meaning you want. You want the points A and B to be the same on the two lines, and then conclude they are actually the same line.Originally Posted by OReilly
You in effect rearranged I.3 like this:
If a "line that contains two distinct points", then "there exists at most one."
Or as you put it "every line that has two distinct points is unique."
But that rearranging changes the meaning of the sentence and is not valid mathematical logic.
So to get I.3 to say what you want, you have to rewrite it, which similar to topsquark could be done like this:
I.3': Two lines that contain the same two distinct points are equal.
But as originally written, I.3 does not say this.