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Metamath Proof Explorer

Theorem cbvralsvw

Description: Change bound variable by using a substitution. Version of cbvralsv with a disjoint variable condition, which does not require ax-13 . (Contributed by NM, 20-Nov-2005) Avoid ax-13 . (Revised by GG, 10-Jan-2024) (Proof shortened by Wolf Lammen, 8-Mar-2025) Avoid ax-10 , ax-12 . (Revised by SN, 21-Aug-2025)

Ref Expression
Assertion cbvralsvw ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 )

Proof

Step Hyp Ref Expression
1 sb8v ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
2 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
3 df-ral ⊢ ( ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) )
4 eleq1w ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
5 4 imbi1d ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 → 𝜑 ) ) )
6 5 sbbiiev ⊢ ( [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ [ 𝑦 / 𝑥 ] ( 𝑦 ∈ 𝐴 → 𝜑 ) )
7 sbrimvw ⊢ ( [ 𝑦 / 𝑥 ] ( 𝑦 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) )
8 6 7 bitr2i ⊢ ( ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
9 8 albii ⊢ ( ∀ 𝑦 ( 𝑦 ∈ 𝐴 → [ 𝑦 / 𝑥 ] 𝜑 ) ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
10 3 9 bitri ⊢ ( ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 ↔ ∀ 𝑦 [ 𝑦 / 𝑥 ] ( 𝑥 ∈ 𝐴 → 𝜑 ) )
11 1 2 10 3bitr4i ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 [ 𝑦 / 𝑥 ] 𝜑 )