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Metamath Proof Explorer

Theorem ralidm

Description: Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997) Reduce axiom usage. (Revised by GG, 2-Sep-2024)

Ref Expression
Assertion ralidm ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step Hyp Ref Expression
1 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )
2 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
3 ax-1 ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
4 3 axc4i ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
5 pm2.21 ⊢ ( ¬ 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
6 sp ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
7 5 6 ja ⊢ ( ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
8 7 alimi ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
9 4 8 impbii ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
10 2 bicomi ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )
11 10 imbi2i ⊢ ( ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )
12 11 albii ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )
13 2 9 12 3bitrri ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ∈ 𝐴 𝜑 ) ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )
14 1 13 bitri ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )