Theorem hbab1OLD

Description: Obsolete version of hbab1 as of 25-Oct-2024. (Contributed by NM, 26-May-1993) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	hbab1OLD	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } → ∀ 𝑥 𝑦 ∈ { 𝑥 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		df-clab	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } ↔ [ 𝑦 / 𝑥 ] 𝜑 )
2		hbs1	⊢ ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑥 [ 𝑦 / 𝑥 ] 𝜑 )
3	1 2	hbxfrbi	⊢ ( 𝑦 ∈ { 𝑥 ∣ 𝜑 } → ∀ 𝑥 𝑦 ∈ { 𝑥 ∣ 𝜑 } )