Theorem 19.21t

Description: Closed form of Theorem 19.21 of Margaris p. 90, see 19.21 . (Contributed by NM, 27-May-1997) (Revised by Mario Carneiro, 24-Sep-2016) (Proof shortened by Wolf Lammen, 3-Jan-2018) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by BJ, 3-Nov-2021)

	Ref	Expression
Assertion	19.21t	$$⊢Ⅎx\phi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		19.38a	$$⊢Ⅎx\phi \to \left(\left(\exists x\phi \to \forall x\psi \right)\leftrightarrow \forall x\left(\phi \to \psi \right)\right)$$
2		19.9t	$$⊢Ⅎx\phi \to \left(\exists x\phi \leftrightarrow \phi \right)$$
3	2	imbi1d	$$⊢Ⅎx\phi \to \left(\left(\exists x\phi \to \forall x\psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)\right)$$
4	1 3	bitr3d	$$⊢Ⅎx\phi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)\right)$$