Theorem 19.23t

Description: Closed form of Theorem 19.23 of Margaris p. 90. See 19.23 . (Contributed by NM, 7-Nov-2005) (Proof shortened by Wolf Lammen, 13-Aug-2020) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by BJ, 8-Oct-2022)

	Ref	Expression
Assertion	19.23t	$$⊢Ⅎx\psi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		19.38b	$$⊢Ⅎx\psi \to \left(\left(\exists x\phi \to \forall x\psi \right)\leftrightarrow \forall x\left(\phi \to \psi \right)\right)$$
2		19.3t	$$⊢Ⅎx\psi \to \left(\forall x\psi \leftrightarrow \psi \right)$$
3	2	imbi2d	$$⊢Ⅎx\psi \to \left(\left(\exists x\phi \to \forall x\psi \right)\leftrightarrow \left(\exists x\phi \to \psi \right)\right)$$
4	1 3	bitr3d	$$⊢Ⅎx\psi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \psi \right)\right)$$