I was asked to apply universal elimination on $\forall x A(x) \rightarrow B,$ but the $\forall$ here is not the main operator, what should I do? here's the question and what I have for now enter image description here

  • $\begingroup$ You cannot..... $\endgroup$ – Mauro ALLEGRANZA Mar 1 at 7:32
  • $\begingroup$ I am trying to prove $\forall x A(x) \rightarrow B \vdash \exists x (A(x) \rightarrow B)$, I will edit the question and post what I have for now $\endgroup$ – n.y Mar 1 at 7:40

Universal elimination when $∀$ is not the main operator.

You cannot apply UE when the quantifier is not the main operator.

You have to derive $\forall x A(x)$ in order to use it to "detach" $B$ using ($\to$-E):

  1. $\forall x A(x) \to B$ --- premise

  2. $\lnot \exists x (A(x) \to B)$ --- assumed [a]

  3. $\lnot A(y)$ --- assumed [b]

  4. $A(y)$ --- assumed [c]

  5. $\bot$

  6. $B$ --- from 5)

  7. $A(y) \to B$ --- from 4) and 6) by ($\to$-I), discharging [c]

  8. $\exists x (A(x) \to B)$ --- from 7) by ($\exists$-I)

  9. $\bot$

  10. $\lnot \lnot A(y)$ --- from 3) and 9) by ($\to$-I), discharging [b]

  11. $A(y)$ --- from 10) by ($\lnot \lnot$-E)

  12. $\forall x A(x)$ --- from 11) by ($\forall$-I)

  13. $B$ --- from 1) and 12) by ($\to$-E)

  14. $A(y) \to B$ --- from 13) by ($\to$-I)

  15. $\exists x (A(x) \to B)$ --- from 14) by ($\exists$-I)

  16. $\bot$

  1. $\exists x (A(x) \to B)$ --- from 2) and 16) by ($\to$-I), discharging [a], followed by ($\lnot \lnot$-E)
  • $\begingroup$ line 14 won't work because $\rightarrow I$ rule needs to start with an assumption, I'll update what I have $\endgroup$ – n.y Mar 1 at 21:20
  • $\begingroup$ @n.yc - it works. It is enough to reiterate the assumption... $\endgroup$ – Mauro ALLEGRANZA Mar 2 at 6:51
  • $\begingroup$ Well it doesn't work as I posted...Did I do something wrong? $\endgroup$ – n.y Mar 3 at 5:06
  • $\begingroup$ @n.y - 10 is not an assumption. After 12 assume $A(a)$ and then re-iterate $B$. $\endgroup$ – Mauro ALLEGRANZA Mar 3 at 6:55
  • $\begingroup$ But in this case the $A(a)\rightarrow B$ will be the second layer, but the final conclusion will need to be at the out most layer? $\endgroup$ – n.y Mar 3 at 21:52

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