how is ∀x (~P(x) or ∃y (P(y) and x ≠ y)) equivalent to
∀x ~P(x) or ∃x ∃y (P(x) and P(y) and x ≠ y)

Question

how is ∀x (~P(x) or ∃y (P(y) and x ≠ y)) equivalent to 
∀x ~P(x) or ∃x ∃y (P(x) and P(y) and x ≠ y)

lyrae · Answer

If you look at it from a pure logical standpoint

we have three variables in our first expression: 
~a or (b and c)

the second statment can be expressed as
~a or (a and b and c)

If you set a = false the first expression always evaluates to true.
If you set a = true the first expression will depend on the values of b and c.

If you set a = false the second expression always evaluates to true.
If you set a = true then the added "a and" won't matter because a already is true and the expression will still depend on b and c.

Or am I missing something?