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Metamath Proof Explorer

Theorem nfabd

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker nfabdw when possible. (Contributed by Mario Carneiro, 8-Oct-2016) Avoid ax-9 and ax-ext . (Revised by Wolf Lammen, 23-May-2023) (New usage is discouraged.)

	Ref	Expression
Hypotheses	nfabd.1	⊢ Ⅎ 𝑦 𝜑
	nfabd.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
Assertion	nfabd	⊢ ( 𝜑 → Ⅎ 𝑥 { 𝑦 ∣ 𝜓 } )

Proof

Step	Hyp	Ref	Expression
1		nfabd.1	⊢ Ⅎ 𝑦 𝜑
2		nfabd.2	⊢ ( 𝜑 → Ⅎ 𝑥 𝜓 )
3		nfv	⊢ Ⅎ 𝑧 𝜑
4		df-clab	⊢ ( 𝑧 ∈ { 𝑦 ∣ 𝜓 } ↔ [ 𝑧 / 𝑦 ] 𝜓 )
5	1 2	nfsbd	⊢ ( 𝜑 → Ⅎ 𝑥 [ 𝑧 / 𝑦 ] 𝜓 )
6	4 5	nfxfrd	⊢ ( 𝜑 → Ⅎ 𝑥 𝑧 ∈ { 𝑦 ∣ 𝜓 } )
7	3 6	nfcd	⊢ ( 𝜑 → Ⅎ 𝑥 { 𝑦 ∣ 𝜓 } )