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Metamath Proof Explorer

Theorem hbab

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by NM, 1-Mar-1995) Add disjoint variable condition to avoid ax-13 . See hbabg for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024)

		Ref	Expression
	Hypothesis	hbab.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
	Assertion	hbab	⊢ ( 𝑧 ∈ { 𝑦 ∣ 𝜑 } → ∀ 𝑥 𝑧 ∈ { 𝑦 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		hbab.1	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
2		df-clab	⊢ ( 𝑧 ∈ { 𝑦 ∣ 𝜑 } ↔ [ 𝑧 / 𝑦 ] 𝜑 )
3	1	hbsbw	⊢ ( [ 𝑧 / 𝑦 ] 𝜑 → ∀ 𝑥 [ 𝑧 / 𝑦 ] 𝜑 )
4	2 3	hbxfrbi	⊢ ( 𝑧 ∈ { 𝑦 ∣ 𝜑 } → ∀ 𝑥 𝑧 ∈ { 𝑦 ∣ 𝜑 } )