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Metamath Proof Explorer

Theorem nfsabg

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 . See nfsab for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by Mario Carneiro, 11-Aug-2016) (New usage is discouraged.)

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

Proof

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