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	|- F/ x ph
	nfim.2	|- F/ x ps
Assertion	nfim	|- F/ x ( ph -> ps )

Proof

Step	Hyp	Ref	Expression
1		nfim.1	|- F/ x ph
2		nfim.2	|- F/ x ps
3		nfimt	|- ( ( F/ x ph /\ F/ x ps ) -> F/ x ( ph -> ps ) )
4	1 2 3	mp2an	|- F/ x ( ph -> ps )