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Mirrors > Home > ILE Home > Th. List > vtocl3ga | GIF version |
Description: Implicit substitution of 3 classes for 3 setvar variables. (Contributed by NM, 20-Aug-1995.) |
Ref | Expression |
---|---|
vtocl3ga.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
vtocl3ga.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
vtocl3ga.3 | ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) |
vtocl3ga.4 | ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) |
Ref | Expression |
---|---|
vtocl3ga | ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2178 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2178 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2178 | . 2 ⊢ Ⅎ𝑧𝐴 | |
4 | nfcv 2178 | . 2 ⊢ Ⅎ𝑦𝐵 | |
5 | nfcv 2178 | . 2 ⊢ Ⅎ𝑧𝐵 | |
6 | nfcv 2178 | . 2 ⊢ Ⅎ𝑧𝐶 | |
7 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜓 | |
8 | nfv 1421 | . 2 ⊢ Ⅎ𝑦𝜒 | |
9 | nfv 1421 | . 2 ⊢ Ⅎ𝑧𝜃 | |
10 | vtocl3ga.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
11 | vtocl3ga.2 | . 2 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
12 | vtocl3ga.3 | . 2 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) | |
13 | vtocl3ga.4 | . 2 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆) → 𝜑) | |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | vtocl3gaf 2622 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 |
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: preq12bg 3544 pocl 4040 sowlin 4057 |
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