Theorem nfimd

Description: If in a context x is not free in ps and ch , then it is not free in ( ps -> ch ) . Deduction form of nfim . (Contributed by Mario Carneiro, 24-Sep-2016) (Proof shortened by Wolf Lammen, 30-Dec-2017) df-nf changed. (Revised by Wolf Lammen, 18-Sep-2021) Eliminate curried form of nfimt . (Revised by Wolf Lammen, 10-Jul-2022)

	Ref	Expression
Hypotheses	nfimd.1	$$⊢\phi \to Ⅎx\psi$$
	nfimd.2	$$⊢\phi \to Ⅎx\chi$$
Assertion	nfimd	$$⊢\phi \to Ⅎx\left(\psi \to \chi \right)$$

Proof

Step	Hyp	Ref	Expression
1		nfimd.1	$$⊢\phi \to Ⅎx\psi$$
2		nfimd.2	$$⊢\phi \to Ⅎx\chi$$
3		19.35	$$⊢\exists x\left(\psi \to \chi \right)\leftrightarrow \left(\forall x\psi \to \exists x\chi \right)$$
4	3	biimpi	$$⊢\exists x\left(\psi \to \chi \right)\to \left(\forall x\psi \to \exists x\chi \right)$$
5	1	nfrd	$$⊢\phi \to \left(\exists x\psi \to \forall x\psi \right)$$
6	2	nfrd	$$⊢\phi \to \left(\exists x\chi \to \forall x\chi \right)$$
7	5 6	imim12d	$$⊢\phi \to \left(\left(\forall x\psi \to \exists x\chi \right)\to \left(\exists x\psi \to \forall x\chi \right)\right)$$
8		19.38	$$⊢\left(\exists x\psi \to \forall x\chi \right)\to \forall x\left(\psi \to \chi \right)$$
9	4 7 8	syl56	$$⊢\phi \to \left(\exists x\left(\psi \to \chi \right)\to \forall x\left(\psi \to \chi \right)\right)$$
10	9	nfd	$$⊢\phi \to Ⅎx\left(\psi \to \chi \right)$$