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 Apr 23 comment Satisfiability proof of formulas with pure literals Hi thx for your reply. Yes, I understand the principle, but I have problems in formulating a precise (structural) induction proof. Apr 23 comment Satisfiability proof of formulas with pure literals Thx for the comment, I've edited my question and included the case for $\psi := \varphi \land \ell$. Is this going in the right direction? Apr 22 comment Validity of a first-order formula yay! Can I ask you yet another question, just to be sure: If I replace $r$ in $\varphi$ by $\doteq$ (equality), so now I am considering also the theory of equality and its axioms, then the formula becomes valid. So, each instance of $r(x,y)$ gets replaced by $x \doteq y$. My assumption is that, then the formula is valid. because $x$ and $y$ are equal and according to the predicate substitution axioms of equality theory $p(x)$ and $p(y)$ always evaluates to the same truth value. Is this kind of the right (formal) argument? Apr 22 comment Validity of a first-order formula Thank you for your reply. So, I choose $\mathbb N_0$ as my Domain. $I(x) = 0$, $I(y)=1$. Moreover the meaning of predicate $p$ is "is even". The meaning of the predicate $r(x,y)$ is $y$ is greater than $x$, and I am done? Jan 29 comment Get rid of an existential quantifier yeah thank you. i've edited my question