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