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Metamath Proof Explorer

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