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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 2175 | . 2 ⊢ ℲyA | |
2 | nfcv 2175 | . 2 ⊢ ℲxB | |
3 | 1, 2 | ralcomf 2465 | 1 ⊢ (∀x ∈ A ∀y ∈ B φ ↔ ∀y ∈ B ∀x ∈ A φ) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∀wral 2300 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 |
This theorem is referenced by: ralcom4 2570 ssint 3622 issod 4047 reusv3 4158 cnvpom 4803 cnvsom 4804 fununi 4910 isocnv2 5395 dfsmo2 5843 |
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