1. $\displaystyle \vdash \exists y \forall x R(x,y) \to \forall x \exists yR(x,y)$

2. $\displaystyle \exists y \forall x R(x,y) \vdash \forall x \exists yR(x,y)$ (by

deduction theorem)

3. $\displaystyle \forall x R(x, c) \vdash \forall x \exists yR(x,y)$ (by rule EI, existential instantiation)

4. $\displaystyle \forall x R(x, c) \vdash \exists yR(x,y)$ (by

generalization theorem)

5. $\displaystyle R(x, c) \vdash \exists yR(x,y)$ (by generalization theorem)

6. $\displaystyle \forall y \neg R(x,y) \vdash \neg R(x,c)$ (contraposition)

7. $\displaystyle \neg R(x,c) \vdash \neg R(x,c)$ (

quantifier axiom)

Thus, (2) is tautology.