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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 2312 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2312 | . 2 ⊢ Ⅎ𝑦𝐴 | |
3 | nfcv 2312 | . 2 ⊢ Ⅎ𝑧𝐴 | |
4 | nfcv 2312 | . 2 ⊢ Ⅎ𝑦𝐵 | |
5 | nfcv 2312 | . 2 ⊢ Ⅎ𝑧𝐵 | |
6 | nfcv 2312 | . 2 ⊢ Ⅎ𝑧𝐶 | |
7 | nfv 1521 | . 2 ⊢ Ⅎ𝑥𝜓 | |
8 | nfv 1521 | . 2 ⊢ Ⅎ𝑦𝜒 | |
9 | nfv 1521 | . 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 2799 | 1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 |
This theorem is referenced by: preq12bg 3760 pocl 4288 sowlin 4305 |
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