![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > optocl | GIF version |
Description: Implicit substitution of class for ordered pair. (Contributed by NM, 5-Mar-1995.) |
Ref | Expression |
---|---|
optocl.1 | ⊢ 𝐷 = (𝐵 × 𝐶) |
optocl.2 | ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) |
optocl.3 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) |
Ref | Expression |
---|---|
optocl | ⊢ (𝐴 ∈ 𝐷 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp3 4507 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶))) | |
2 | opelxp 4483 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
3 | optocl.3 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) → 𝜑) | |
4 | 2, 3 | sylbi 120 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶) → 𝜑) |
5 | optocl.2 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | syl5ib 153 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶) → 𝜓)) |
7 | 6 | imp 123 | . . . 4 ⊢ ((〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶)) → 𝜓) |
8 | 7 | exlimivv 1825 | . . 3 ⊢ (∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶)) → 𝜓) |
9 | 1, 8 | sylbi 120 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝜓) |
10 | optocl.1 | . 2 ⊢ 𝐷 = (𝐵 × 𝐶) | |
11 | 9, 10 | eleq2s 2183 | 1 ⊢ (𝐴 ∈ 𝐷 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ∃wex 1427 ∈ wcel 1439 〈cop 3455 × cxp 4452 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2624 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-opab 3908 df-xp 4460 |
This theorem is referenced by: 2optocl 4530 3optocl 4531 ecoptocl 6395 ax1rid 7475 ax0id 7476 axcnre 7479 |
Copyright terms: Public domain | W3C validator |