The Logic Book, M. Bergmann, J. Moor, J. Nelson, McGraw Hill 2004, 10.5E, 4(b), p. 563

We have to show that the following pair of sentences are equivalent. I may well have done this example before but I haven't been keeping a very scrupulous record of the answers I have already posted. At any rate, two sentences are equivalent if they can each be derived from the other. Thus:

1. (∃x)(∃y)Axy ⊃Aab
2. (∃x)(∃y)Axy ≡ Aab
3. * (∃x)(∃y)Axy ......... ACP
4. * Aab ......... 3,1MP
5. (∃x)(∃y)Axy ⊃Aab ......... 3-4CP
6. * Aab ......... ACP
7. * (∃y)Aay ......... 6EG
8. * (∃x)(∃y)Axy ......... 7EG
9. Aab ⊃(∃x)(∃y)Axy ......... 6-8CP
10. [(∃x)(∃y)Axy ⊃Aab] • [Aab ⊃(∃x)(∃y)Axy] ......... 5,9Conj.
11. (∃x)(∃y)Axy ≡ Aab ......... 10BE

This way we have derived (∃x)(∃y)Axy ≡ Aab from (∃x)(∃y)Axy ⊃Aab. To go in the opposite direction it is enough to break down (∃x)(∃y)Axy ≡ Aab into a conjunction of two conditional sentences and simplify by dropping Aab ⊃(∃x)(∃y)Axy. We'll be left with (∃x)(∃y)Axy ⊃Aab.