Suppose you have four real numbers $a,b,c,d$ and you know that $a=b$ and $c=d$.
Use the above four axioms to deduce that $a+d=b+c$.
The four axioms of equality are
- Reflexivity. $x=x$
- Symmetry. $x=y\implies y=x$
- Transitivity. $x=y\land y=z\implies x=z$
- Substitution. $x=y\implies f(x)=f(y)$
Since $c=d$, by the axiom of symmetry, we have $d=c$. Let's create a function $f(x)=a+x$. By axiom of substitution, since $d=c$, we have$f(d)=f(c)$, that is, $a+d=a+c$.
Let's create another function $g(x)=x+c$. By the axiom of substitution, since $a=b$, we have $f(a)=f(b)$, that is, $a+c=b+c$.
By the axiom of transitivity, $a+d=a+c$ and $a+c=b+c$ gives us $a+d=b+c$, the desired conclusion.
