Theorem nfbr

Description: Bound-variable hypothesis builder for binary relation. (Contributed by NM, 1-Sep-1999) (Revised by Mario Carneiro, 14-Oct-2016)

	Ref	Expression
Hypotheses	nfbr.1	⊢ Ⅎ 𝑥 𝐴
	nfbr.2	⊢ Ⅎ 𝑥 𝑅
	nfbr.3	⊢ Ⅎ 𝑥 𝐵
Assertion	nfbr	⊢ Ⅎ 𝑥 𝐴 𝑅 𝐵

Proof

Step	Hyp	Ref	Expression
1		nfbr.1	⊢ Ⅎ 𝑥 𝐴
2		nfbr.2	⊢ Ⅎ 𝑥 𝑅
3		nfbr.3	⊢ Ⅎ 𝑥 𝐵
4	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
5	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝑅 )
6	3	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐵 )
7	4 5 6	nfbrd	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 𝑅 𝐵 )
8	7	mptru	⊢ Ⅎ 𝑥 𝐴 𝑅 𝐵