Theorem r19.23v

Description: Restricted quantifier version of 19.23v . Version of r19.23 with a disjoint variable condition. (Contributed by NM, 31-Aug-1999) Reduce dependencies on axioms. (Revised by Wolf Lammen, 14-Jan-2020)

	Ref	Expression
Assertion	r19.23v	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\in A\phi \to \psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		con34b	$$⊢\left(\phi \to \psi \right)\leftrightarrow \left(\neg \psi \to \neg \phi \right)$$
2	1	ralbii	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \forall x\in A\left(\neg \psi \to \neg \phi \right)$$
3		r19.21v	$$⊢\forall x\in A\left(\neg \psi \to \neg \phi \right)\leftrightarrow \left(\neg \psi \to \forall x\in A\neg \phi \right)$$
4		dfrex2	$$⊢\exists x\in A\phi \leftrightarrow \neg \forall x\in A\neg \phi$$
5	4	imbi1i	$$⊢\left(\exists x\in A\phi \to \psi \right)\leftrightarrow \left(\neg \forall x\in A\neg \phi \to \psi \right)$$
6		con1b	$$⊢\left(\neg \forall x\in A\neg \phi \to \psi \right)\leftrightarrow \left(\neg \psi \to \forall x\in A\neg \phi \right)$$
7	5 6	bitr2i	$$⊢\left(\neg \psi \to \forall x\in A\neg \phi \right)\leftrightarrow \left(\exists x\in A\phi \to \psi \right)$$
8	2 3 7	3bitri	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\in A\phi \to \psi \right)$$