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Mirrors > Home > ILE Home > Th. List > rr19.3v | GIF version |
Description: Restricted quantifier version of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 25-Oct-2012.) |
Ref | Expression |
---|---|
rr19.3v | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd 171 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ 𝜑)) | |
2 | 1 | rspcv 2785 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 𝜑 → 𝜑)) |
3 | 2 | ralimia 2493 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜑) |
4 | ax-1 6 | . . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝐴 → 𝜑)) | |
5 | 4 | ralrimiv 2504 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 𝜑) |
6 | 5 | ralimi 2495 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑) |
7 | 3, 6 | impbii 125 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 ∀wral 2416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 |
This theorem is referenced by: (None) |
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