Theorem bj-nfimt

Description: Closed form of nfim and curried (exported) form of nfimt . (Contributed by BJ, 20-Oct-2021)

	Ref	Expression
Assertion	bj-nfimt	$$⊢Ⅎx\phi \to \left(Ⅎx\psi \to Ⅎx\left(\phi \to \psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		19.35	$$⊢\exists x\left(\phi \to \psi \right)\leftrightarrow \left(\forall x\phi \to \exists x\psi \right)$$
2		id	$$⊢Ⅎx\phi \to Ⅎx\phi$$
3	2	nfrd	$$⊢Ⅎx\phi \to \left(\exists x\phi \to \forall x\phi \right)$$
4	3	imim1d	$$⊢Ⅎx\phi \to \left(\left(\forall x\phi \to \exists x\psi \right)\to \left(\exists x\phi \to \exists x\psi \right)\right)$$
5	1 4	biimtrid	$$⊢Ⅎx\phi \to \left(\exists x\left(\phi \to \psi \right)\to \left(\exists x\phi \to \exists x\psi \right)\right)$$
6		id	$$⊢Ⅎx\psi \to Ⅎx\psi$$
7	6	nfrd	$$⊢Ⅎx\psi \to \left(\exists x\psi \to \forall x\psi \right)$$
8	7	imim2d	$$⊢Ⅎx\psi \to \left(\left(\exists x\phi \to \exists x\psi \right)\to \left(\exists x\phi \to \forall x\psi \right)\right)$$
9		19.38	$$⊢\left(\exists x\phi \to \forall x\psi \right)\to \forall x\left(\phi \to \psi \right)$$
10	8 9	syl6	$$⊢Ⅎx\psi \to \left(\left(\exists x\phi \to \exists x\psi \right)\to \forall x\left(\phi \to \psi \right)\right)$$
11	5 10	syl9	$$⊢Ⅎx\phi \to \left(Ⅎx\psi \to \left(\exists x\left(\phi \to \psi \right)\to \forall x\left(\phi \to \psi \right)\right)\right)$$
12		df-nf	$$⊢Ⅎx\left(\phi \to \psi \right)\leftrightarrow \left(\exists x\left(\phi \to \psi \right)\to \forall x\left(\phi \to \psi \right)\right)$$
13	11 12	imbitrrdi	$$⊢Ⅎx\phi \to \left(Ⅎx\psi \to Ⅎx\left(\phi \to \psi \right)\right)$$