Let Fa be a $ \mathcal L_1^{=} $ - closed formula and $ \Delta$ = { $\forall x(Fx \to Gx), \lnot Ga$} a set of $ \mathcal L_1^{=} $ - closed formulas, where F and G are predicate symbols and $\mathbf a$ is an constant. According to the classic semantics, Fa is the semantic consequence of $\Delta $ ie $ \Delta \vDash$ Fa?
What have I done so far
By contradiction
Suppose $ \Delta $ is not a semantic consequence of Fa. Then there is B (Fa) = false,. By definition B ($ \forall x (Fx \to Gx) $) = true and B ($ \lnot $ Ga) = true. But if B ($ \forall (Fi \to Gi) $) = T, this implies that B ($ \forall (Fa \to Ga) $)= V then or B (Fa) = F or B (Ga) = V, but by the hypothesis B (Fa) = F, so B (Ga) needs to be either T or F, as we assume that B ($ \lnot $ Ga) = T. No contradiction.