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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