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Mirrors > Home > ILE Home > Th. List > ralidm | GIF version |
Description: Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.) |
Ref | Expression |
---|---|
ralidm | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra1 2521 | . . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 | |
2 | anidm 396 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴) | |
3 | rsp2 2540 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑)) | |
4 | 2, 3 | biimtrrid 153 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑)) |
5 | 1, 4 | ralrimi 2561 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
6 | ax-1 6 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) | |
7 | nfra1 2521 | . . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
8 | 7 | 19.23 1689 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) |
9 | 6, 8 | sylibr 134 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) |
10 | df-ral 2473 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) | |
11 | 9, 10 | sylibr 134 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑) |
12 | 5, 11 | impbii 126 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1362 ∃wex 1503 ∈ wcel 2160 ∀wral 2468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-ral 2473 |
This theorem is referenced by: issref 5029 cnvpom 5189 |
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