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)

	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		bi2.04	$$⊢\left(x\in A\to \left(\phi \to \psi \right)\right)\leftrightarrow \left(\phi \to \left(x\in A\to \psi \right)\right)$$
2	1	albii	$$⊢\forall x\left(x\in A\to \left(\phi \to \psi \right)\right)\leftrightarrow \forall x\left(\phi \to \left(x\in A\to \psi \right)\right)$$
3		19.21v	$$⊢\forall x\left(\phi \to \left(x\in A\to \psi \right)\right)\leftrightarrow \left(\phi \to \forall x\left(x\in A\to \psi \right)\right)$$
4	2 3	bitri	$$⊢\forall x\left(x\in A\to \left(\phi \to \psi \right)\right)\leftrightarrow \left(\phi \to \forall x\left(x\in A\to \psi \right)\right)$$
5		df-ral	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \forall x\left(x\in A\to \left(\phi \to \psi \right)\right)$$
6		df-ral	$$⊢\forall x\in A\psi \leftrightarrow \forall x\left(x\in A\to \psi \right)$$
7	6	imbi2i	$$⊢\left(\phi \to \forall x\in A\psi \right)\leftrightarrow \left(\phi \to \forall x\left(x\in A\to \psi \right)\right)$$
8	4 5 7	3bitr4i	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\in A\psi \right)$$