# quantifier restrictions in natural deduction

Universal elimination and existential introduction are easy. But when it comes to existential elimination and universal introduction, there are some restrictions which I don't fully understand.

Let me give an example:

Show that if ∀x(P(x) → Q(x)) and ∃xQ(x) → R(y) are true, then ∀x(P(x) → R(y)) is true by natural deduction.

1. $P(t)$ --assumed
2. $P(t) \rightarrow Q(t)$ -- universal elimination from the first premise
3. $Q(t)$-- modus ponens 1,2

Now I want to make use of the second premise:

1. $Q(t) \rightarrow R(y)$ -- assumed

I am not sure but I think I cannot do that since the constant $t$ has been used before. Should I give it another letter and proceed like this?

1. $Q(w) \rightarrow R(y)$ -- assumed
2. $R(y)$ -- modus ponens 3, 4(can I do this?)
3. $R(y)$ -- existential elimination (second premise, 4, 5)
4. $P(t) \rightarrow R(y)$ -- implication introduction
5. universal introduction of 7

Can somebody please explain how to go before/after the line 4 so that existential elimination and universal generalization wouldn't be a problem?

## EDIT:

After thinking and thinking I have come up with this

$\\1.\forall x(P(x)\rightarrow Q(x))$ - premise

$\\2. \exists xQ(x) \rightarrow R(y)$ -premise

$\\3.\forall zP(z)$ -assumed

$\\4.Q(a)\rightarrow R(y)$ -assumed

$\\5.P(a)\rightarrow Q(a)$ -universal\ elimination from \1

$\\6. P(a)$ -universal\ elimination\ from\ 3

$\\7. Q(a)$ -modus ponens\ 5,6

$\\8. R(y)$-modus ponens \ 4,7

$\\9. R(y)$-existential \ elimination \ 2,4,8

$\\10. \forall z P(z) \rightarrow R(y)$ -implication \ introduction\ 3, 9

$\\11. P(a)$ - assumed

$\\12. P(a) \rightarrow R(y)$- universal\ elimination from\ 10

$\\13. R(y)$- modus ponens\ 11,12

$\\14. P(a) \rightarrow R(y)$-implication\ introduction\ 11, 13

$\\15. \forall x(P(x) \rightarrow R(y))$- universal introduction from\ 14

Can anybody tell me if this contains any errors?

• Wait... ∃xQ(x) → R(y) isn't clear. Does that mean ((∃xQ(x)) → R(y)) or does it mean (∃x(Q(x)→ R(y))? Oct 24, 2015 at 22:06
• @DougSpoonwood the existential quantifier only affects the first term, Q(x), not the whole compound proposition. So it is the former in your sentence. Oct 24, 2015 at 22:08
• The EDIT part is wrong; see at Doug's answer below. You cannot apply $\exists$-elim that way in step 9. Oct 25, 2015 at 7:25

You get to Q(t) in step 3. So, just use existential introduction to infer ∃xQ(x). Then infer R(y) by modus ponens. Then us conditional introduction to get to (P(t) $\rightarrow$ R(y)).