This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ralidmw

Description: Idempotent law for restricted quantifier. Weak version of ralidm , which does not require ax-10 , ax-12 , but requires ax-8 . (Contributed by GG, 30-Sep-2024)

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

Proof

Step Hyp Ref Expression
1 ralidmw.1 ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
2 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
3 2 imbi2i ⊢ ( ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ∈ 𝐴 𝜑 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
4 3 albii ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ∈ 𝐴 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
5 pm2.21 ⊢ ( ¬ 𝑥 ∈ 𝐴 → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
6 eleq1w ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴 ) )
7 6 1 imbi12d ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∈ 𝐴 → 𝜑 ) ↔ ( 𝑦 ∈ 𝐴 → 𝜓 ) ) )
8 7 spw ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
9 5 8 ja ⊢ ( ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
10 9 alimi ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
11 7 hba1w ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ∀ 𝑥 ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
12 ax-1 ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
13 11 12 alrimih ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) )
14 10 13 impbii ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
15 4 14 bitri ⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ∈ 𝐴 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜑 ) )
16 df-ral ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑥 ∈ 𝐴 𝜑 ) )
17 15 16 2 3bitr4i ⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 𝜑 ↔ ∀ 𝑥 ∈ 𝐴 𝜑 )