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Metamath Proof Explorer

Theorem cbvralvw

Description: Change the bound variable of a restricted universal quantifier using implicit substitution. Version of cbvralv with a disjoint variable condition, which does not require ax-10 , ax-11 , ax-12 , ax-13 . (Contributed by NM, 28-Jan-1997) Avoid ax-13 . (Revised by GG, 10-Jan-2024)

Ref Expression
Hypothesis cbvralvw.1 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion cbvralvw ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 𝜓 )

Proof

Step Hyp Ref Expression
1 cbvralvw.1 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2 eleq1w ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
3 2 1 imbi12d ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 → 𝜓 ) ) )
4 3 cbvalvw ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝜓 ) )
5 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
6 df-ral ⊢ ( ∀ 𝑦 ∈ 𝐴 𝜓 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝐴 → 𝜓 ) )
7 4 5 6 3bitr4i ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑦 ∈ 𝐴 𝜓 )