Theorem 19.38b

Description: Under a non-freeness hypothesis, the implication 19.38 can be strengthened to an equivalence. See also 19.38a . (Contributed by BJ, 3-Nov-2021) (Proof shortened by Wolf Lammen, 9-Jul-2022)

	Ref	Expression
Assertion	19.38b	$$⊢Ⅎx\psi \to \left(\left(\exists x\phi \to \forall x\psi \right)\leftrightarrow \forall x\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		19.38	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)$$
2		exim	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \exists x\psi \right)$$
3		id	$$⊢Ⅎx\psi \to Ⅎx\psi$$
4	3	nfrd	$$⊢Ⅎx\psi \to \left(\exists x\psi \to \forall x\psi \right)$$
5	2 4	syl9r	$$⊢Ⅎx\psi \to \left(\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$
6	1 5	impbid2	$$⊢Ⅎx\psi \to \left(\left(\exists x\phi \to \forall x\psi \right)\leftrightarrow \forall x\left(\phi \to \psi \right)\right)$$