A stands for full-name quantifier and e stands for existential quantifier.

Let F(x):x be a rational number, G(x):x be an integer, and H(x):x be a real number.

Premises: Ax(F(x)→G(x)), Ex(F(x)∧H(x))

Conclusion: Ex(H(x)∧G(x))

Prove:

( 1)Ex(F(x)∧H(x))

(2)F(c)∧H(c) ( 1)EI

(3)F(c)

(4)H(c)

⑤Ax(F(x)→G(x))

(6)F(c)→G(c) (5)UI

(7)G(c) (3)(6) Hypothetical reasoning

(8)G(c)∧H(c)

(9)Ex(H(x)∧G(x)) (8)EG