Metamath Proof Explorer


Theorem hbsbw

Description: If z is not free in ph , it is not free in [ y / x ] ph when y and z are distinct. Version of hbsb with a disjoint variable condition, which requires fewer axioms. (Contributed by NM, 12-Aug-1993) (Revised by Gino Giotto, 23-May-2024) (Proof shortened by Wolf Lammen, 14-May-2025)

Ref Expression
Hypothesis hbsbw.1 ( 𝜑 → ∀ 𝑧 𝜑 )
Assertion hbsbw ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 hbsbw.1 ( 𝜑 → ∀ 𝑧 𝜑 )
2 1 sbimi ( [ 𝑦 / 𝑥 ] 𝜑 → [ 𝑦 / 𝑥 ] ∀ 𝑧 𝜑 )
3 sbal ( [ 𝑦 / 𝑥 ] ∀ 𝑧 𝜑 ↔ ∀ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )
4 2 3 sylib ( [ 𝑦 / 𝑥 ] 𝜑 → ∀ 𝑧 [ 𝑦 / 𝑥 ] 𝜑 )