(adsbygoogle = window.adsbygoogle || []).push({}); Propositional Logic - "There Exists" dual

1. The problem statement, all variables and given/known data

"State and prove the [itex]\exists[/itex]-dual of 6.1.8"

Section 6.1.8 shows how to change [itex]\vdash (\forall x)(\forall y)A \equiv (\forall y)(\forall x)A[/itex] via this method:

(Line 1) [itex](\forall x)(\forall y)A[/itex] (Hypothesis)

(Line 2) [itex](\forall y)A[/itex] (Line 1 + spec)

(Line 3) [itex]A[/itex] (Line 2 + spec)

(Line 4) [itex](\forall x)A[/itex] (Line 3 + gen; Line 1 has no free x)

(Line 5) [itex](\forall y)(\forall x)A[/itex] (Line 4 + gen; Line 1 has no free y)

The [itex]\exists[/itex]-dual of this would be to show [itex]\vdash (\exists x)(\exists y)A \equiv (\exists y)(\exists x)A[/itex].

2. Relevant equations

A crucial formula here is that [itex](\forall x)A \equiv \neg (\exists x) \neg A[/itex]

3. The attempt at a solution

I potentially solved it, but something seems off.

This is what I did:

[itex]\vdash (\exists x)(\exists y)A \equiv (\exists y)(\exists x)A[/itex]

(Line 1) [itex](\exists x)(\exists y)A[/itex]

(Line 2) [itex]\neg (\forall x)\neg (\exists y)A[/itex] Definition of [itex]\exists[/itex]

(Line 3) [itex]\neg (\forall x)\neg \neg (\forall y)\neg A[/itex] Definition of [itex]\exists[/itex]

(Line 4) [itex]\neg (\forall x)(\forall y)\neg A[/itex] Double negative

(Line 5) [itex]\neg (\forall y)\neg A[/itex] Spec

(Line 6) [itex]\neg \neg A[/itex] Spec

(Line 7) [itex]\neg (\forall x)\neg A[/itex] Gen

(Line 8) [itex]\neg (\forall y)(\forall x)\neg A[/itex] Gen

(Line 9) [itex]\neg (\forall y)\neg \neg (\forall x)\neg A[/itex] Double negative

(Line 10) [itex]\neg (\forall y)\neg (\exists x)A[/itex] Definition of [itex]\forall[/itex]

(Line 11) [itex](\exists y)(\exists x)A[/itex] Definition of [itex]\forall[/itex]

What worries me is going from Line 5 to 6, what happens to the first [itex]\neg[/itex] symbol? Can I just leave it there, and then insert my [itex](\forall x)[/itex] between the two [itex]\neg[/itex]s in Line 7 like that?

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# Homework Help: Propositional Logic - There Exists dual

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