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Metamath Proof Explorer

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	⊢ Ⅎ 𝑥 𝜑
	nfim.2	⊢ Ⅎ 𝑥 𝜓
Assertion	nfim	⊢ Ⅎ 𝑥 ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		nfim.1	⊢ Ⅎ 𝑥 𝜑
2		nfim.2	⊢ Ⅎ 𝑥 𝜓
3		nfimt	⊢ ( ( Ⅎ 𝑥 𝜑 ∧ Ⅎ 𝑥 𝜓 ) → Ⅎ 𝑥 ( 𝜑 → 𝜓 ) )
4	1 2 3	mp2an	⊢ Ⅎ 𝑥 ( 𝜑 → 𝜓 )