Theorem bj-19.23t

Description: Statement 19.23t proved from modalK (obsoleting 19.23v ). (Contributed by BJ, 2-Dec-2023)

	Ref	Expression
Assertion	bj-19.23t	$$⊢Ⅎ\text{'}x\psi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		bj-nnf-exlim	$$⊢Ⅎ\text{'}x\psi \to \left(\forall x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \psi \right)\right)$$
2		bj-nnfa	$$⊢Ⅎ\text{'}x\psi \to \left(\psi \to \forall x\psi \right)$$
3	2	imim2d	$$⊢Ⅎ\text{'}x\psi \to \left(\left(\exists x\phi \to \psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$
4		19.38	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)$$
5	3 4	syl6	$$⊢Ⅎ\text{'}x\psi \to \left(\left(\exists x\phi \to \psi \right)\to \forall x\left(\phi \to \psi \right)\right)$$
6	1 5	impbid	$$⊢Ⅎ\text{'}x\psi \to \left(\forall x\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\phi \to \psi \right)\right)$$