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Metamath Proof Explorer

Theorem hblem

Description: Change the free variable of a hypothesis builder. (Contributed by NM, 21-Jun-1993) (Revised by Andrew Salmon, 11-Jul-2011) Add disjoint variable condition to avoid ax-13 . See hblemg for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024)

Ref Expression
Hypothesis hblem.1 ⊢ y ∈ A → ∀ x y ∈ A
Assertion hblem ⊢ z ∈ A → ∀ x z ∈ A

Proof

Step Hyp Ref Expression
1 hblem.1 ⊢ y ∈ A → ∀ x y ∈ A
2 1 hbsbw ⊢ z y y ∈ A → ∀ x z y y ∈ A
3 clelsb1 ⊢ z y y ∈ A ↔ z ∈ A
4 3 albii ⊢ ∀ x z y y ∈ A ↔ ∀ x z ∈ A
5 2 3 4 3imtr3i ⊢ z ∈ A → ∀ x z ∈ A