Theorem reximdai

Description: Deduction from Theorem 19.22 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 31-Aug-1999)

	Ref	Expression
Hypotheses	reximdai.1	$$⊢Ⅎx\phi$$
	reximdai.2	$$⊢\phi \to \left(x\in A\to \left(\psi \to \chi \right)\right)$$
Assertion	reximdai	$$⊢\phi \to \left(\exists x\in A\psi \to \exists x\in A\chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		reximdai.1	$$⊢Ⅎx\phi$$
2		reximdai.2	$$⊢\phi \to \left(x\in A\to \left(\psi \to \chi \right)\right)$$
3	1 2	ralrimi	$$⊢\phi \to \forall x\in A\left(\psi \to \chi \right)$$
4		rexim	$$⊢\forall x\in A\left(\psi \to \chi \right)\to \left(\exists x\in A\psi \to \exists x\in A\chi \right)$$
5	3 4	syl	$$⊢\phi \to \left(\exists x\in A\psi \to \exists x\in A\chi \right)$$