Theorem rexim

Description: Theorem 19.22 of Margaris p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994) (Proof shortened by Andrew Salmon, 30-May-2011)

	Ref	Expression
Assertion	rexim	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\exists x\in A\phi \to \exists x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		con3	$$⊢\left(\phi \to \psi \right)\to \left(\neg \psi \to \neg \phi \right)$$
2	1	ral2imi	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\forall x\in A\neg \psi \to \forall x\in A\neg \phi \right)$$
3	2	con3d	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\neg \forall x\in A\neg \phi \to \neg \forall x\in A\neg \psi \right)$$
4		dfrex2	$$⊢\exists x\in A\phi \leftrightarrow \neg \forall x\in A\neg \phi$$
5		dfrex2	$$⊢\exists x\in A\psi \leftrightarrow \neg \forall x\in A\neg \psi$$
6	3 4 5	3imtr4g	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\exists x\in A\phi \to \exists x\in A\psi \right)$$