Theorem bj-19.21t

Description: Statement 19.21t proved from modalK (obsoleting 19.21v ). (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Assertion	bj-19.21t	$$⊢Ⅎ\text{'}x\phi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-alrim	$$⊢Ⅎ\text{'}x\phi \to \left(\forall x\left(\phi \to \psi \right)\to \left(\phi \to \forall x\psi \right)\right)$$
2		bj-nnfe	$$⊢Ⅎ\text{'}x\phi \to \left(\exists x\phi \to \phi \right)$$
3	2	imim1d	$$⊢Ⅎ\text{'}x\phi \to \left(\left(\phi \to \forall x\psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$
4		19.38	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)$$
5	3 4	syl6	$$⊢Ⅎ\text{'}x\phi \to \left(\left(\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)\right)$$
6	1 5	impbid	$$⊢Ⅎ\text{'}x\phi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\phi \to \forall x\psi \right)\right)$$