Assuming I have the statement ∀x(∀y¬Q(x,y)∨P(x)), can I pull the universal quantifier ∀y out of the parenthesis? Meaning, is this statement equivalent to ∀x∀y(¬Q(x,y)∨P(x)) ?
An approach I tried so far:
- ∀x((∃y Q(x,y) ) => P(x)). (original eq.)
- ∀x((∀y¬Q(x,y))∨P(x)) (De Morgan's application)
- ∀x∀y(¬Q(x,y)∨P(x)). (Working off the assumption that taking out the ∀y is a valid operation).
- ∀x∀y (Q(x,y) => P(x)) (Going backwards from the ¬P v Q definition of implication)
Statement 4 does not seem to be equivalent to statement 1, which suggests that pulling out the universal quantifier is not acceptable. I would greatly appreciate any confirmation of whether this is the case, and if so, what governs when quantifiers can be brought to the outside of the parenthesis.