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Mirrors > Home > ILE Home > Th. List > rspc2v | GIF version |
Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.) |
Ref | Expression |
---|---|
rspc2v.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
rspc2v.2 | ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspc2v | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1461 | . 2 ⊢ Ⅎ𝑥𝜒 | |
2 | nfv 1461 | . 2 ⊢ Ⅎ𝑦𝜓 | |
3 | rspc2v.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
4 | rspc2v.2 | . 2 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) | |
5 | 1, 2, 3, 4 | rspc2 2711 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 ∀wral 2348 |
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 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 |
This theorem is referenced by: rspc2va 2714 rspc3v 2716 wetriext 4319 f1veqaeq 5429 isorel 5468 fovcl 5626 caovclg 5673 caovcomg 5676 smoel 5938 supmoti 6406 supsnti 6418 isotilem 6419 cauappcvgprlem1 6849 caucvgprlemnkj 6856 caucvgprlemnbj 6857 caucvgprprlemval 6878 frecuzrdgrrn 9410 iseqcaopr3 9460 iseqhomo 9468 climcn2 10148 |
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