$$xA $$x$$A$,$ for all formulas A$.$

$$xA $$x$$A$,$ for all formulas A$.$

$($xA $$xB$)$x$($A $$ B$),$ for all formulas A$,$ B$.$

$$x$($A $$ B$)($xA $$xB$),$ for all formulas A$,$ B$.$

$$x$($A $$ B$)|=($xA $$xB$),$ for all formulas A$,$ B$.$

NOT $($xA $$xB$)|=$x$($A $$ B$)$ in general

$$x$$yA $$y$$xA

$$x$$yA $$y$$xA

$$y$$xA $|=$x$$yA$,$ for all formulas A$.$

NOT $$x$$yA $|=$y$$xA in general

$$xA $|=$xA$,$ because of our requirement that every universe M must be nonempty.

$$x$$y$($x $=$ y $$ fx $=$ fy$)$ is valid.

$$x$$y$($fx $=$ fy $$ x $=$ y$)$ is NOT valid.

Exercise $2$ Verify each line in item $3$ above. For the two lines beginning NOT give specific

formulas A $($and B$)$ for which the relation is false, and show it is false by giving a specific

structure which satisfies the left hand side but not the right hand side. For the last line, give

a structure which does not satisfy the formula.