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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 2732 | . . 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 2317 | . . . . 5 ⊢ Ⅎ𝑦 𝐴 ∈ V |
7 | vtocl3gf.2 | . . . . 5 ⊢ Ⅎ𝑦𝜒 | |
8 | 6, 7 | nfim 1559 | . . . 4 ⊢ Ⅎ𝑦(𝐴 ∈ V → 𝜒) |
9 | vtocl3gf.c | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
10 | 9 | nfel1 2317 | . . . . 5 ⊢ Ⅎ𝑧 𝐴 ∈ V |
11 | vtocl3gf.3 | . . . . 5 ⊢ Ⅎ𝑧𝜃 | |
12 | 10, 11 | nfim 1559 | . . . 4 ⊢ Ⅎ𝑧(𝐴 ∈ V → 𝜃) |
13 | vtocl3gf.5 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
14 | 13 | imbi2d 229 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒))) |
15 | vtocl3gf.6 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | |
16 | 15 | imbi2d 229 | . . . 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 2779 | . . . 4 ⊢ (𝐴 ∈ V → 𝜓) |
22 | 2, 3, 4, 8, 12, 14, 16, 21 | vtocl2gf 2783 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ V → 𝜃)) |
23 | 1, 22 | mpan9 279 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → 𝜃) |
24 | 23 | 3impb 1188 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 967 = wceq 1342 Ⅎwnf 1447 ∈ wcel 2135 Ⅎwnfc 2293 Vcvv 2721 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 |
This theorem is referenced by: vtocl3gaf 2790 |
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