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Metamath Proof Explorer

Theorem nfsbcw

Description: Bound-variable hypothesis builder for class substitution. Version of nfsbc with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 7-Sep-2014) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

	Ref	Expression
Hypotheses	nfsbcw.1	⊢ Ⅎ 𝑥 𝐴
	nfsbcw.2	⊢ Ⅎ 𝑥 𝜑
Assertion	nfsbcw	⊢ Ⅎ 𝑥 [ 𝐴 / 𝑦 ] 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfsbcw.1	⊢ Ⅎ 𝑥 𝐴
2		nfsbcw.2	⊢ Ⅎ 𝑥 𝜑
3		nftru	⊢ Ⅎ 𝑦 ⊤
4	1	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝐴 )
5	2	a1i	⊢ ( ⊤ → Ⅎ 𝑥 𝜑 )
6	3 4 5	nfsbcdw	⊢ ( ⊤ → Ⅎ 𝑥 [ 𝐴 / 𝑦 ] 𝜑 )
7	6	mptru	⊢ Ⅎ 𝑥 [ 𝐴 / 𝑦 ] 𝜑