  S-metric and b-metric spaces are metrizable, but it is still quite impossible
to get an explicit form of the concerned metric function. To overcome this, the
notion of $\phi$-metric is developed by making a suitable modification in
triangle inequality and its properties are pretty similar to metric function.
It is shown that one can easily construct a $\phi$-metric from existing
generalized distance functions like S-metric, b-metric, etc. and those are
$\phi$-metrizable. The convergence of sequence on those metric spaces is
identical to the respective induced $\phi$-metric spaces. So, unlike metrics,
concerned $\phi$-metric can be easily constructed and $\phi$-metric functions
may play the role of metric functions substantially. Also, the structure of
$\phi$-metric spaces is studied and some fixed point theorems are established.
