Negating a Math Equation: ( \exists x ) ( \forall y ) \Phi (x,y )

[gnome]

I want to negate this: ( \exists x ) ( \forall y ) \Phi (x,y )

Is this correct?

\neg ( ( \exists x ) ( \forall y ) \Phi (x,y ) ) \equiv ( \forall x ) ( \exists y ) \neg \Phi (x,y )

 

[AKG]

looks right

 

[Owen Holden]

I want to negate this: ( \exists x ) ( \forall y ) \Phi (x,y )

Is this correct?

\neg ( ( \exists x ) ( \forall y ) \Phi (x,y ) ) \equiv ( \forall x ) ( \exists y ) \neg \Phi (x,y )

Yes.

~[(Ex)(Ay)F(x,y)] <-> ~(Ex)(Ay)F(x.y)
~(Ex)(Ay)F(x,y) <-> (Ax)~(Ay)F(x,y)
(Ax)~(Ay)F(x,y) <-> (Ax)(Ey)~F(x,y)
therefore,
~[(Ex)(Ay)F(x,y)] <-> (Ax)(Ey)~F(x,y).

 

[gnome]

Thanks guys.