I think there’s a logical error. You claim to be deducing “IF route FAST is taken THEN I will arrive at 3pm”, but what you should actually be deducing is “IF route SLOW is taken THEN (IF route FAST is taken THEN I will arrive at 3pm)”. What you end up with is proving that “route SLOW is taken” is logically equivalent to “IF route FAST is taken THEN I will arrive at 3pm AND IF route SLOW is taken THEN I will arrive at 2pm”, but neither of them are proved.
I think there’s a logical error. You claim to be deducing “IF route FAST is taken THEN I will arrive at 3pm”, but what you should actually be deducing is “IF route SLOW is taken THEN (IF route FAST is taken THEN I will arrive at 3pm)”. What you end up with is proving that “route SLOW is taken” is logically equivalent to “IF route FAST is taken THEN I will arrive at 3pm AND IF route SLOW is taken THEN I will arrive at 2pm”, but neither of them are proved.