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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 2409 | . . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 | |
2 | anidm 388 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴) | |
3 | rsp2 2425 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → 𝜑)) | |
4 | 2, 3 | syl5bir 151 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → (𝑥 ∈ 𝐴 → 𝜑)) |
5 | 1, 4 | ralrimi 2444 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
6 | ax-1 5 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) | |
7 | nfra1 2409 | . . . . 5 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝜑 | |
8 | 7 | 19.23 1613 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) |
9 | 6, 8 | sylibr 132 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) |
10 | df-ral 2364 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑥 ∈ 𝐴 𝜑)) | |
11 | 9, 10 | sylibr 132 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑) |
12 | 5, 11 | impbii 124 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1287 ∃wex 1426 ∈ wcel 1438 ∀wral 2359 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-4 1445 ax-ial 1472 ax-i5r 1473 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-ral 2364 |
This theorem is referenced by: issref 4814 cnvpom 4973 |
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