Theorem reximia

Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 10-Feb-1997) (Proof shortened by Wolf Lammen, 31-Oct-2024)

	Ref	Expression
Hypothesis	reximia.1	$$⊢x\in A\to \left(\phi \to \psi \right)$$
Assertion	reximia	$$⊢\exists x\in A\phi \to \exists x\in A\psi$$

Proof

Step	Hyp	Ref	Expression
1		reximia.1	$$⊢x\in A\to \left(\phi \to \psi \right)$$
2	1	imdistani	$$⊢\left(x\in A\wedge \phi \right)\to \left(x\in A\wedge \psi \right)$$
3	2	reximi2	$$⊢\exists x\in A\phi \to \exists x\in A\psi$$