Exercise $3$

Show that, for $|--{?}_N($FOL$)$:

$\frac{|--{?}_{N}AB}{,\text{A}\text{A}\text{x}\text{.}A|--{?}_N\text{A}\text{A}\text{x}\text{.}B}$

is admissible in $N$ with no side conditions.

$\frac{|--{?}_N\text{A}\text{A}\text{x}\text{.}A}{|--{?}_N{A}_t^x}$

is admissible for legal $A_t^x.$

This version of the deduction theorem and its inverse:

$\frac{,A|--{?}_{N}B}{|--{?}_N\text{A}\text{A}\text{A}B},\frac{|--{?}_N\text{A}\text{A}\text{A}B}{,A|--{?}_{N}B}$