Theorem r19.21t

Description: Restricted quantifier version of 19.21t ; closed form of r19.21 . (Contributed by NM, 1-Mar-2008) (Proof shortened by Wolf Lammen, 2-Jan-2020)

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

Proof

Step	Hyp	Ref	Expression
1		19.21t	$$⊢Ⅎx\phi \to \left(\forall x\left(\phi \to \left(x\in A\to \psi \right)\right)\leftrightarrow \left(\phi \to \forall x\left(x\in A\to \psi \right)\right)\right)$$
2		df-ral	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \forall x\left(x\in A\to \left(\phi \to \psi \right)\right)$$
3		bi2.04	$$⊢\left(x\in A\to \left(\phi \to \psi \right)\right)\leftrightarrow \left(\phi \to \left(x\in A\to \psi \right)\right)$$
4	3	albii	$$⊢\forall x\left(x\in A\to \left(\phi \to \psi \right)\right)\leftrightarrow \forall x\left(\phi \to \left(x\in A\to \psi \right)\right)$$
5	2 4	bitri	$$⊢\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \forall x\left(\phi \to \left(x\in A\to \psi \right)\right)$$
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	1 5 7	3bitr4g	$$⊢Ⅎx\phi \to \left(\forall x\in A\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\in A\psi \right)\right)$$