Consider the following premises. For all x_10, if x_10 were a carpenter, then x_10 is not climbing a mountain. For all x_10, x_10 will climb a mountain. For all x_10, x_10 is happy if and only if x_10 is a politician. There is at least one x_10 for which x_10 is a carpenter if and only if x_10 is happy. Can we infer the following from them? If we can, name the inference rule being used: For all x_10, if x_10 were a carpenter, then x_10 is not climbing a mountain.
That is one of the premises. Therefore, the answer is yes.