As I know, and ordered pair can be defined in terms of sets, so equality of ordered pairs is the same as equality of sets. And I found such a theorem on the Wikipedia page about ordered pairs
$(a, b) = (c, d)$ if and only if $a = c$ and $b = d$.
And my question is this, what do this equalities $$a=c$$ and $$b=d$$ mean in terms of logic and set theory ?
I know that equality can be defined as a relation, so does it mean this relation there ?
First, the definition of $=$(for sets, not proper class) is(usually): $(A=B)\iff(\forall x(x\in A\iff x\in B))$
The usual definition of ordered pair is: $(a,b)\iff\{\{a\},\{a,b\}\}$
If you want to prove that $(a,b)=(c,d)\iff a=c\land b=d$ using the definition of equality I would suggest to first prove it using $=$ and then replace all of the $=$ with the definition.
Notice $a,b,c,d$ has to be sets with this definition, otherwise it is undefined