Theorem hbabg

Description: Bound-variable hypothesis builder for a class abstraction. Usage of this theorem is discouraged because it depends on ax-13 . See hbab for a version with more disjoint variable conditions, but not requiring ax-13 . (Contributed by NM, 1-Mar-1995) (New usage is discouraged.)

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

Proof

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