Theorem nfcsbw

Description: Bound-variable hypothesis builder for substitution into a class. Version of nfcsb with a disjoint variable condition, which does not require ax-13 . (Contributed by Mario Carneiro, 12-Oct-2016) Avoid ax-13 . (Revised by Gino Giotto, 10-Jan-2024)

	Ref	Expression
Hypotheses	nfcsbw.1	⊢ Ⅎ 𝑥 𝐴
	nfcsbw.2	⊢ Ⅎ 𝑥 𝐵
Assertion	nfcsbw	⊢ Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵

Proof

Step	Hyp	Ref	Expression
1		nfcsbw.1	⊢ Ⅎ 𝑥 𝐴
2		nfcsbw.2	⊢ Ⅎ 𝑥 𝐵
3		df-csb	⊢ ⦋ 𝐴 / 𝑦 ⦌ 𝐵 = { 𝑧 ∣ [ 𝐴 / 𝑦 ] 𝑧 ∈ 𝐵 }
4		nftru	⊢ Ⅎ 𝑧 ⊤
5		nftru	⊢ Ⅎ 𝑦 ⊤
6	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
7	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐵 )
8	7	nfcrd	⊢ ( ⊤ → Ⅎ 𝑥 𝑧 ∈ 𝐵 )
9	5 6 8	nfsbcdw	⊢ ( ⊤ → Ⅎ 𝑥 [ 𝐴 / 𝑦 ] 𝑧 ∈ 𝐵 )
10	4 9	nfabdw	⊢ ( ⊤ → Ⅎ 𝑥 { 𝑧 ∣ [ 𝐴 / 𝑦 ] 𝑧 ∈ 𝐵 } )
11	3 10	nfcxfrd	⊢ ( ⊤ → Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵 )
12	11	mptru	⊢ Ⅎ 𝑥 ⦋ 𝐴 / 𝑦 ⦌ 𝐵