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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 2312 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2312 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | ralcomf 2631 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wral 2448 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 |
This theorem is referenced by: ralcom4 2752 ssint 3847 issod 4304 reusv3 4445 cnvpom 5153 cnvsom 5154 fununi 5266 isocnv2 5791 dfsmo2 6266 ixpiinm 6702 rexfiuz 10953 tgss2 12873 cnmptcom 13092 |
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