Theorem 19.38

Description: Theorem 19.38 of Margaris p. 90. The converse holds under non-freeness conditions, see 19.38a and 19.38b . (Contributed by NM, 12-Mar-1993) Allow a shortening of 19.21t . (Revised by Wolf Lammen, 2-Jan-2018)

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

Proof

Step	Hyp	Ref	Expression
1		alnex	$$⊢\forall x\neg \phi \leftrightarrow \neg \exists x\phi$$
2		pm2.21	$$⊢\neg \phi \to \left(\phi \to \psi \right)$$
3	2	alimi	$$⊢\forall x\neg \phi \to \forall x\left(\phi \to \psi \right)$$
4	1 3	sylbir	$$⊢\neg \exists x\phi \to \forall x\left(\phi \to \psi \right)$$
5		ala1	$$⊢\forall x\psi \to \forall x\left(\phi \to \psi \right)$$
6	4 5	ja	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)$$