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Dynamical phase transition (DPT) characterizes the abrupt change of dynamical properties in nonequilibrium quantum many-body systems. It has been demonstrated that extra quantum fluctuating modes besides the conventional order parameter field can drastically change the properties of equilibrium phase transitions. However, the counterpart phenomena in DPTs have rarely been explored. Here, we study the DPT in the Dirac system after a sudden quench, and find that the fermion fluctuations can round a putative first-order DPT into a dynamical critical point, which is referred to as a fermion-induced dynamical critical point (FIDCP). It is also a nonthermal critical point, in which the universal short-time scaling behavior emerges despite the system goes through a first-order transition after thermalization. In the novel scenario of FIDCP, the quantum Yukawa coupling $g_q$ is indispensable for inducing the FIDCP albeit irrelevant in the infrared scale. We call these variables {\it indispensable irrelevant scaling variables}. Moreover, a dynamical tricritical point which separates the first-order DPT and the FIDCP is discovered by tuning this indispensable irrelevant scaling variable. We further mention possible experimental realizations.