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Mirrors > Home > ILE Home > Th. List > 2optocl | GIF version |
Description: Implicit substitution of classes for ordered pairs. (Contributed by NM, 12-Mar-1995.) |
Ref | Expression |
---|---|
2optocl.1 | ⊢ 𝑅 = (𝐶 × 𝐷) |
2optocl.2 | ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) |
2optocl.3 | ⊢ (〈𝑧, 𝑤〉 = 𝐵 → (𝜓 ↔ 𝜒)) |
2optocl.4 | ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑) |
Ref | Expression |
---|---|
2optocl | ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2optocl.1 | . . 3 ⊢ 𝑅 = (𝐶 × 𝐷) | |
2 | 2optocl.3 | . . . 4 ⊢ (〈𝑧, 𝑤〉 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | 2 | imbi2d 229 | . . 3 ⊢ (〈𝑧, 𝑤〉 = 𝐵 → ((𝐴 ∈ 𝑅 → 𝜓) ↔ (𝐴 ∈ 𝑅 → 𝜒))) |
4 | 2optocl.2 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | imbi2d 229 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜑) ↔ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜓))) |
6 | 2optocl.4 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ (𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷)) → 𝜑) | |
7 | 6 | ex 114 | . . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜑)) |
8 | 1, 5, 7 | optocl 4527 | . . . 4 ⊢ (𝐴 ∈ 𝑅 → ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜓)) |
9 | 8 | com12 30 | . . 3 ⊢ ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → (𝐴 ∈ 𝑅 → 𝜓)) |
10 | 1, 3, 9 | optocl 4527 | . 2 ⊢ (𝐵 ∈ 𝑅 → (𝐴 ∈ 𝑅 → 𝜒)) |
11 | 10 | impcom 124 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ∈ wcel 1439 〈cop 3453 × cxp 4450 |
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 3963 ax-pow 4015 ax-pr 4045 |
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 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-opab 3906 df-xp 4458 |
This theorem is referenced by: 3optocl 4529 ecopovsym 6402 ecopovsymg 6405 th3qlem2 6409 axaddcom 7466 |
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