Theorem r19.2z

Description: Theorem 19.2 of Margaris p. 89 with restricted quantifiers (compare 19.2 ). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003)

	Ref	Expression
Assertion	r19.2z	$$⊢\left(A\ne \varnothing \wedge \forall x\in A\phi \right)\to \exists x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
2		exintr	$$⊢\forall x\left(x\in A\to \phi \right)\to \left(\exists xx\in A\to \exists x\left(x\in A\wedge \phi \right)\right)$$
3	1 2	sylbi	$$⊢\forall x\in A\phi \to \left(\exists xx\in A\to \exists x\left(x\in A\wedge \phi \right)\right)$$
4		n0	$$⊢A\ne \varnothing \leftrightarrow \exists xx\in A$$
5		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
6	3 4 5	3imtr4g	$$⊢\forall x\in A\phi \to \left(A\ne \varnothing \to \exists x\in A\phi \right)$$
7	6	impcom	$$⊢\left(A\ne \varnothing \wedge \forall x\in A\phi \right)\to \exists x\in A\phi$$