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Mirrors > Home > ILE Home > Th. List > vtocl3gf | GIF version |
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.) |
Ref | Expression |
---|---|
vtocl3gf.a | ⊢ Ⅎ𝑥𝐴 |
vtocl3gf.b | ⊢ Ⅎ𝑦𝐴 |
vtocl3gf.c | ⊢ Ⅎ𝑧𝐴 |
vtocl3gf.d | ⊢ Ⅎ𝑦𝐵 |
vtocl3gf.e | ⊢ Ⅎ𝑧𝐵 |
vtocl3gf.f | ⊢ Ⅎ𝑧𝐶 |
vtocl3gf.1 | ⊢ Ⅎ𝑥𝜓 |
vtocl3gf.2 | ⊢ Ⅎ𝑦𝜒 |
vtocl3gf.3 | ⊢ Ⅎ𝑧𝜃 |
vtocl3gf.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl3gf.5 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl3gf.6 | ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) |
vtocl3gf.7 | ⊢ 𝜑 |
Ref | Expression |
---|---|
vtocl3gf | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2566 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | vtocl3gf.d | . . . 4 ⊢ Ⅎ𝑦𝐵 | |
3 | vtocl3gf.e | . . . 4 ⊢ Ⅎ𝑧𝐵 | |
4 | vtocl3gf.f | . . . 4 ⊢ Ⅎ𝑧𝐶 | |
5 | vtocl3gf.b | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
6 | 5 | nfel1 2188 | . . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ V |
7 | vtocl3gf.2 | . . . . 5 ⊢ Ⅎ𝑦𝜒 | |
8 | 6, 7 | nfim 1464 | . . . 4 ⊢ Ⅎ𝑦(𝐴 ∈ V → 𝜒) |
9 | vtocl3gf.c | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
10 | 9 | nfel1 2188 | . . . . 5 ⊢ Ⅎ𝑧 𝐴 ∈ V |
11 | vtocl3gf.3 | . . . . 5 ⊢ Ⅎ𝑧𝜃 | |
12 | 10, 11 | nfim 1464 | . . . 4 ⊢ Ⅎ𝑧(𝐴 ∈ V → 𝜃) |
13 | vtocl3gf.5 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
14 | 13 | imbi2d 219 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒))) |
15 | vtocl3gf.6 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | |
16 | 15 | imbi2d 219 | . . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴 ∈ V → 𝜒) ↔ (𝐴 ∈ V → 𝜃))) |
17 | vtocl3gf.a | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
18 | vtocl3gf.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
19 | vtocl3gf.4 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
20 | vtocl3gf.7 | . . . . 5 ⊢ 𝜑 | |
21 | 17, 18, 19, 20 | vtoclgf 2612 | . . . 4 ⊢ (𝐴 ∈ V → 𝜓) |
22 | 2, 3, 4, 8, 12, 14, 16, 21 | vtocl2gf 2615 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ V → 𝜃)) |
23 | 1, 22 | mpan9 265 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → 𝜃) |
24 | 23 | 3impb 1100 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 885 = wceq 1243 Ⅎwnf 1349 ∈ wcel 1393 Ⅎwnfc 2165 Vcvv 2557 |
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-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 |
This theorem is referenced by: vtocl3gaf 2622 |
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