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Mirrors > Home > ILE Home > Th. List > 3optocl | GIF version |
Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.) |
Ref | Expression |
---|---|
3optocl.1 | ⊢ 𝑅 = (𝐷 × 𝐹) |
3optocl.2 | ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) |
3optocl.3 | ⊢ (〈𝑧, 𝑤〉 = 𝐵 → (𝜓 ↔ 𝜒)) |
3optocl.4 | ⊢ (〈𝑣, 𝑢〉 = 𝐶 → (𝜒 ↔ 𝜃)) |
3optocl.5 | ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐹) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐹) ∧ (𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹)) → 𝜑) |
Ref | Expression |
---|---|
3optocl | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑅) → 𝜃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3optocl.1 | . . . 4 ⊢ 𝑅 = (𝐷 × 𝐹) | |
2 | 3optocl.4 | . . . . 5 ⊢ (〈𝑣, 𝑢〉 = 𝐶 → (𝜒 ↔ 𝜃)) | |
3 | 2 | imbi2d 219 | . . . 4 ⊢ (〈𝑣, 𝑢〉 = 𝐶 → (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜃))) |
4 | 3optocl.2 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | imbi2d 219 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜑) ↔ ((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜓))) |
6 | 3optocl.3 | . . . . . . 7 ⊢ (〈𝑧, 𝑤〉 = 𝐵 → (𝜓 ↔ 𝜒)) | |
7 | 6 | imbi2d 219 | . . . . . 6 ⊢ (〈𝑧, 𝑤〉 = 𝐵 → (((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜓) ↔ ((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜒))) |
8 | 3optocl.5 | . . . . . . 7 ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐹) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐹) ∧ (𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹)) → 𝜑) | |
9 | 8 | 3expia 1106 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐹) ∧ (𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐹)) → ((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜑)) |
10 | 1, 5, 7, 9 | 2optocl 4417 | . . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → ((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → 𝜒)) |
11 | 10 | com12 27 | . . . 4 ⊢ ((𝑣 ∈ 𝐷 ∧ 𝑢 ∈ 𝐹) → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒)) |
12 | 1, 3, 11 | optocl 4416 | . . 3 ⊢ (𝐶 ∈ 𝑅 → ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜃)) |
13 | 12 | impcom 116 | . 2 ⊢ (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) ∧ 𝐶 ∈ 𝑅) → 𝜃) |
14 | 13 | 3impa 1099 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑅) → 𝜃) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 〈cop 3378 × cxp 4343 |
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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
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-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-xp 4351 |
This theorem is referenced by: ecopovtrn 6203 ecopovtrng 6206 |
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