Theorem nfim

Description: If x is not free in ph and ps , then it is not free in ( ph -> ps ) . Inference associated with nfimt . (Contributed by Mario Carneiro, 11-Aug-2016) (Proof shortened by Wolf Lammen, 2-Jan-2018) df-nf changed. (Revised by Wolf Lammen, 17-Sep-2021)

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

Proof

Step	Hyp	Ref	Expression
1		nfim.1	$$⊢Ⅎx\phi$$
2		nfim.2	$$⊢Ⅎx\psi$$
3		nfimt	$$⊢\left(Ⅎx\phi \wedge Ⅎx\psi \right)\to Ⅎx\left(\phi \to \psi \right)$$
4	1 2 3	mp2an	$$⊢Ⅎx\left(\phi \to \psi \right)$$