I'm stucking in understanding the usage and soundness of the rules for the quantifiers $\forall,\exists$ in sequent calculus.
$\forall-L$: $~~~~~\dfrac{\Gamma,\phi[t]\vdash \Delta}{\Gamma,\forall x\phi[x/t]\vdash\Delta}$ $~~~~~~~~~~~~~~~~~~\forall-R$: $~~~~~\dfrac{\Gamma\vdash\phi[y],\Delta}{\Gamma\vdash\forall x\phi[x/y],\Delta}$
$\exists -L$: $~~~~~\dfrac{\Gamma,\phi[y]\vdash \Delta}{\Gamma,\exists x\phi[x/y]\vdash\Delta}$$~~~~~~~~~~~~~~~~~~\exists-R$: $~~~~~\dfrac{\Gamma\vdash\phi[t],\Delta}{\Gamma\vdash\exists x\phi[x/t],\Delta}$
Take $\forall-L$ for example, how do "$\Gamma,\phi[t]\vdash \Delta$" tautologically imply "$\Gamma,\forall x\phi[x/t]\vdash\Delta$"?
Also, why is $\exists -L$ sound? If for every model $(\mathfrak{A},\sigma)$ satisfies $\Gamma,\phi[y]$ also satisfies $\Delta$, why $\Gamma,\exists x\phi[x/y]\vDash\Delta$? I tried hard to think about it, but it's so complicated... (I'm not used to the open formula involved in $\vdash$ or $\vDash$ symbol as well.) Detailed answer is welcome. Thanks.