The multiparameter bifurcation problem for \(\lambda=(\lambda_1,\cdots,\lambda_p)\in \mathbb R^p\) given by \[ F(\lambda,x)\equiv Bx-\sum_{i=1}^{p} \lambda_i Ax+N(\lambda,x)=0, \quad N(\lambda,0)=0, \quad D_x N(\lambda,0)=0, \] is considered in real Banach spaces \(X,Y\). In the case that \(\dim N(D_xF(\lambda^0,0))=n\geq 1\) and \(\text{codim}\,R(D_xF(\lambda^0,0))=m\leq p\), the Lyapunov-Schmidt method reduces the original equation to a finite-dimensional bifurcation equation of the form \(f(\lambda,u)=L(\lambda)u+q(\lambda)u+h(\lambda,u)(u):\mathbb R^p\times \mathbb R^n\rightarrow \mathbb R^m\). As a generalization of the simple eigenvalue hypothesis [\textit{M. G. Crandall} and \textit{P. Rabinowitz}, J. Funct. Anal. 8, 321--340 (1971; Zbl 0219.46015)], the condition \[ \text{for all }\phi\in \mathbb R^n \text{ with } \|\phi\|=1, \text{ span}[L_1\phi,\cdots L_p\phi]=\mathbb R^m \tag{\text{C}} \] is introduced and a bifurcation existence theorem together with the genericity of (C) is proved.