Theorem 19.38a

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

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

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		id	$$⊢Ⅎx\phi \to Ⅎx\phi$$
3	2	nfrd	$$⊢Ⅎx\phi \to \left(\exists x\phi \to \forall x\phi \right)$$
4		alim	$$⊢\forall x\left(\phi \to \psi \right)\to \left(\forall x\phi \to \forall x\psi \right)$$
5	3 4	syl9	$$⊢Ⅎx\phi \to \left(\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$
6	1 5	impbid2	$$⊢Ⅎx\phi \to \left(\left(\exists x\phi \to \forall x\psi \right)\leftrightarrow \forall x\left(\phi \to \psi \right)\right)$$