A new approach to generalize metric functions

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.