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Mirrors > Home > ILE Home > Th. List > ralcom | GIF version |
Description: Commutation of restricted quantifiers. (Contributed by NM, 13-Oct-1999.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
ralcom | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2223 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2223 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | ralcomf 2521 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∀wral 2353 |
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-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-nf 1391 df-sb 1688 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 |
This theorem is referenced by: ralcom4 2632 ssint 3678 issod 4109 reusv3 4245 cnvpom 4925 cnvsom 4926 fununi 5033 isocnv2 5529 dfsmo2 5982 rexfiuz 10247 |
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