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Metamath Proof Explorer

Theorem nfaba1

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016) Add disjoint variable condition to avoid ax-13 . See nfaba1g for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024) Avoid ax-12 . (Revised by SN, 14-May-2025)

		Ref	Expression
	Assertion	nfaba1	⊢ Ⅎ 𝑥 { 𝑦 ∣ ∀ 𝑥 𝜑 }

Proof

Step	Hyp	Ref	Expression
1		df-clab	⊢ ( 𝑧 ∈ { 𝑦 ∣ ∀ 𝑥 𝜑 } ↔ [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 )
2		sbal	⊢ ( [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑 ↔ ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 )
3		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑
4	2 3	nfxfr	⊢ Ⅎ 𝑥 [ 𝑧 / 𝑦 ] ∀ 𝑥 𝜑
5	1 4	nfxfr	⊢ Ⅎ 𝑥 𝑧 ∈ { 𝑦 ∣ ∀ 𝑥 𝜑 }
6	5	nfci	⊢ Ⅎ 𝑥 { 𝑦 ∣ ∀ 𝑥 𝜑 }