which of the following are valid propositions? Let P(x) and Q(x) be arbitrary predicates. Which of the following statements is always TRUE?
1.((∀x(P(x) ∨ Q(x)))) ⟹ ((∀xP(x)) ∨ (∀xQ(x)))
2.(∀x(P (x) ⟹ Q (x))) ⟹ ((∀xP(x)) ⟹ (∀xQ(x)))
3.(∀x(P(x)) ⟹ ∀x (Q(x))) ⟹ (∀x(P(x) ⟹ Q(x)))
4.(∀x(P(x)) ⇔ (∀x (Q(x)))) ⟹ (∀x(P(x) ⇔ Q(x)))
How to approach such questions ,I tried with the second option and I couldn't get that how is it valid since for all quantifier is not distributive over disjunction so how can we write ∀x(~P(x)∨Q(x)) as ∀x(~P(x))∨∀x(Q(x))  , so If I try converting from LHS to RHS then I will have to distribute for all quantifier over disjunction which I can't do so how is option 2 valid ?
 A: It's probably better to prove 2 using say, natural deduction, rather than to try to manipulate symbols. 
Intuitively, this is just "All humans are mortals; therefore, if everything is a human, then everything is mortal." If you were to draw a Venn diagram, from $\forall x(P(x) \to Q(x))$, you'd have two circles, with the $P$ circle entirely contained within the $Q$ circle. If $\forall x P(x)$, then everything in the universe is a $P$: there's no gap between the $P$ an $Q$ circles, $P$ fills $Q$. So everything is a $Q$.


*

*$\forall x (P(x) \to Q(x)) \quad\text{assumption}$

*$\forall x P(x)\quad\quad\quad\quad\,\,\text{assumption}$

*$P(a) \quad\quad\quad\quad\quad\text{instantiate $\forall$ in 2}$

*$P(a)\to Q(a) \quad\quad\text{instantiate $\forall$ in 1}$ 

*$Q(a) \quad\quad\quad\quad\quad\text{from 3, 4 by modus ponens}$ 

*$\forall x Q(x) \quad\quad\quad\quad\text{eliminate arbitrary a}$

*$(\forall x P(x) \to \forall xQ(x)) \quad\quad\text{discharge assumption 2}$ 

*$\forall x (P(x) \to Q(x)) \to (\forall x P(x) \to \forall xQ(x)) \quad\text{discharge assumption 1}$

