Example 6

Test for validity ("x)ØF(x,x) ® ($x)("y)ØF(x,y).

          ' 1.        Ø(("x)ØF(x,x) ® ($x)("y)ØF(x,y))          N

a₂ a₁  \ 2.        ("x)ØF(x,x)                                              1

          ' 3.       Ø($x)("y)ØF(x,y)                                      1

a₂ a₁  \ 4.        ("x)Ø("y)ØF(x,y)                                     3, QN

            5.       ØF(a₁,a₁)                                                  2, UI

          ' 6.       Ø("y)ØF(a₁,y)                                          4, UI

     a₂  ' 7.       ($y)ØØF(a₁,y)                                           6, QN

          ' 8.       ØØF(a₁,a₂)                                                7, EI

            9.       F(a₁,a₂)                                                     8

          10.       ØF(a₂,a₂)                                                   2, UI

        ' 11.       Ø("y)ØF(a₂,y)                                           4, UI

    a₃ ' 12.       ($y)ØØF(a₂,y)                                           11, QN

        ' 13.       ØØF(a₂,a₃)                                                12, EI

          14.       F(a₂,a₃)                                                     13

          etc...

This branch will never be closed. It will eventually contain an arbitrarily long sequence of wff F(a_n,a_n+1) and ØF(a_n,a_n). It will have used a sequence of constants a₁, a₂, ... a_n, ...

We construct a model for the test wff's counterexample with domain {a₁, a₂, ... } = | A |

Define F^A such that F^A = {(a_n, a_n+1) | n in the Naturals}

Then |=_A ("x)ØF(x,x) but |¹_A ($x)("y)ØF(x,y)

So |¹_A ("x)ØF(x,x) ® ($x)("y)ØF(x,y)