The article deals with new generalization of coincidence point theorem. It is assumed that $A, B, C, D$ are self-mappings of a metric space $(X,d)$ satisfying, for all $x, y \in X$, the inequality $$F(d(Ax,By),d(Sx,Ty),d(Ax,Sx),d(By,Ty),d(Sx,By),d(Ty,Ax)) \le 0,$$ where $F: {\Bbb R}_+^6 \to {\Bbb R}$ is a function with properties: (F$_1$) $F(t,0,t,0,0,t) > 0$ for all $t > 0$; (F$_2$) $F(t,0,0,t,t,0) > 0$ for all $t > 0$; (F$_3$) $F(t,t,0,0,t,t) > 0$ for all $t > 0$. Furthermore, it is assumed that there exist two sequences $\{x_n\}, \{y_n\}$ such that $$\lim_{n \to \infty} Ax_n = \lim_{n \to \infty} Sx_n = \lim_{n \to \infty} By_n = \lim_{n \to \infty} Ty_n = t \quad \text{for some} \ t \in X,$$ and that $S(X)$ and $T(X)$ are closed. Under these conditions, it is stated that both pairs $(A,S)$ and $(B,T)$ have a coincidence point and, moreover, $A, B, C, T$ have a unique common fixed point, provided that both pairs $(A,S)$ and $(B,T)$ are weakly compatible ($(A,S)$ is called weakly compatible if $Ax = Sx$ for some $X$ implies $ASx = SAX$, and, similarly, for $(B,T)$). Some modifications of this statement are also given. A number of examples of functions $F$ are presented. Comparisons between old and new results are gathered at the end of the article.