Theorem nfss

Description: If x is not free in A and B , it is not free in A C_ B . (Contributed by NM, 27-Dec-1996)

	Ref	Expression
Hypotheses	dfssf.1	⊢ Ⅎ 𝑥 𝐴
	dfssf.2	⊢ Ⅎ 𝑥 𝐵
Assertion	nfss	⊢ Ⅎ 𝑥 𝐴 ⊆ 𝐵

Proof

Step	Hyp	Ref	Expression
1		dfssf.1	⊢ Ⅎ 𝑥 𝐴
2		dfssf.2	⊢ Ⅎ 𝑥 𝐵
3	1 2	dfss3f	⊢ ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 )
4		nfra1	⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 𝑥 ∈ 𝐵
5	3 4	nfxfr	⊢ Ⅎ 𝑥 𝐴 ⊆ 𝐵