Theorem nfsab

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016) Add disjoint variable condition to avoid ax-13 . See nfsabg for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024)

	Ref	Expression
Hypothesis	nfsab.1	⊢ Ⅎ 𝑥 𝜑
Assertion	nfsab	⊢ Ⅎ 𝑥 𝑧 ∈ { 𝑦 ∣ 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		nfsab.1	⊢ Ⅎ 𝑥 𝜑
2	1	nf5ri	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
3	2	hbab	⊢ ( 𝑧 ∈ { 𝑦 ∣ 𝜑 } → ∀ 𝑥 𝑧 ∈ { 𝑦 ∣ 𝜑 } )
4	3	nf5i	⊢ Ⅎ 𝑥 𝑧 ∈ { 𝑦 ∣ 𝜑 }