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Mirrors > Home > ILE Home > Th. List > rspc2 | GIF version |
Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 9-Nov-2012.) |
Ref | Expression |
---|---|
rspc2.1 | ⊢ Ⅎ𝑥𝜒 |
rspc2.2 | ⊢ Ⅎ𝑦𝜓 |
rspc2.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
rspc2.4 | ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspc2 | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2178 | . . . 4 ⊢ Ⅎ𝑥𝐷 | |
2 | rspc2.1 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
3 | 1, 2 | nfralxy 2360 | . . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐷 𝜒 |
4 | rspc2.3 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
5 | 4 | ralbidv 2326 | . . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝐷 𝜑 ↔ ∀𝑦 ∈ 𝐷 𝜒)) |
6 | 3, 5 | rspc 2650 | . 2 ⊢ (𝐴 ∈ 𝐶 → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → ∀𝑦 ∈ 𝐷 𝜒)) |
7 | rspc2.2 | . . 3 ⊢ Ⅎ𝑦𝜓 | |
8 | rspc2.4 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜒 ↔ 𝜓)) | |
9 | 7, 8 | rspc 2650 | . 2 ⊢ (𝐵 ∈ 𝐷 → (∀𝑦 ∈ 𝐷 𝜒 → 𝜓)) |
10 | 6, 9 | sylan9 389 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐷 𝜑 → 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 ∀wral 2306 |
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 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 |
This theorem is referenced by: rspc2v 2662 |
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