Deductive Logic, Warren Goldfarb, Hackett, 2003, Part IV, Ex. 5(c)

The proof of this argument is not particularly challenging but it is notable for the application of the first law of identity which says that everything is identical with itself, and which we can introduce at any point without justification (line 5). Since we are asked to show an equivalence, we have to work both ways.

1. (x)(y)(x = y ⊃Gxy)
2. ∴(x)Gxx
3. (y)(x = y ⊃Gxy) / 1UI x/x
4. x = x ⊃Gxx / 3UI y/x
5. (x)(x = x) / Id
6. x = x / 5UI x/x
7. Gxx / 6,4 MP
8. (x)Gxx / 7UG

1. (x)Gxx
2. ∴(x)(y)(x = y ⊃Gxy)
3. * x = y / ACP
4. * Gxx / 1UI x/x
5. * Gxy / 3,4Id
6. x = y ⊃Gxy / 3-5CP
7. (y)(x = y ⊃Gxy) / 6UG
8. (x)(y)(x = y ⊃Gxy) 7UG