Theorem r19.21v

Description: Restricted quantifier version of 19.21v . (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020) (Proof shortened by Wolf Lammen, 11-Dec-2024)

	Ref	Expression
Assertion	r19.21v	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\in A\psi \right)$$

Proof

Step	Hyp	Ref	Expression
1		pm2.27	$$⊢\phi \to \left(\left(\phi \to \psi \right)\to \psi \right)$$
2	1	ralimdv	$$⊢\phi \to \left(\forall x\in A\left(\phi \to \psi \right)\to \forall x\in A\psi \right)$$
3	2	com12	$$⊢\forall x\in A\left(\phi \to \psi \right)\to \left(\phi \to \forall x\in A\psi \right)$$
4		pm2.21	$$⊢\neg \phi \to \left(\phi \to \psi \right)$$
5	4	ralrimivw	$$⊢\neg \phi \to \forall x\in A\left(\phi \to \psi \right)$$
6		ax-1	$$⊢\psi \to \left(\phi \to \psi \right)$$
7	6	ralimi	$$⊢\forall x\in A\psi \to \forall x\in A\left(\phi \to \psi \right)$$
8	5 7	ja	$$⊢\left(\phi \to \forall x\in A\psi \right)\to \forall x\in A\left(\phi \to \psi \right)$$
9	3 8	impbii	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\in A\psi \right)$$