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I have a habit of trying out correctness about some logical statements with worlfram alpha by generating truth table for them. For example, I can try if this:

$$((¬x→y)∧(¬x→¬y))→x$$

is correct or not by generating truth table for $((¬x→y)∧(¬x→¬y))$ which turns out to be the same as $x$ column in the same truth table. Hence the above is correct. However is there any way I can check same for biconditionals involving nested existential and universal quantifiers and predicates? For example can I somehow verify rules of this kind?:

$$(∀x)(∀y)ϕ(x,y)⇔(∀y)(∀x)ϕ(x,y)$$

Update

I am able to do following check $∀x,y(x∨y)$ as follows:

Resolve[ForAll[{x,y}, x or y]]

which correctly returns $False$ as $(x∨y)$ does not hold for all $x$ and $y$.

So now I thought I can do something similar to obtain $True$ for following (which is a general fact): $¬(∀x)ϕ(x)⇔(∃x)¬ϕ(x)$. I tried this:

Resolve[ForAll[x,(not ForAll[x, x]) xnor (exists[x,not x])]]

But it did not work. Note that $⇔$ is nothing but XNOR. So how do I do this especially something like following also correctly returns $True$:

Resolve[not ForAll[x, x]]

which stands for $¬∀x(x)$.

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    I think you are conflating quantification over a universe with quantification over propositions. In quantifying over a universe, you might say $$\exists x \forall y ~ M(x, y) \implies \forall y \exists x ~ M(x, y)$$ When quantifying over propositions you might say $$\Pi a \Pi b~(a \Rightarrow (b \Rightarrow a))$$ The second can be checked with a truth table, the first cannot, unless you are assuming the universe in the first case is finite. – 2017-01-23
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    **Q1.** What $\Pi$ stands for? **Q2.** Are the two equations above mean the same? Also, I just ran [$\neg\forall x(x)$](http://www.wolframalpha.com/input/?i=Resolve%5Bx,not+ForAll%5Bx,+x%5D%5D) and surprisingly it returned $\exists x(\neg x)$. Now thinking how can I do something similar for $(∃x)(ϕ(x)∧ ψ(x))→((∃x)ϕ(x)∧ (∃x)ψ(x))$. **Q3.** Any thoughts? – 2017-01-23
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    You can do a sort of check with Euler diagrams but they get messy very quickly and proof by picture is always a bad idea. – 2017-01-24

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