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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 4610 | . . . 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 4610 | . 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 1331 ∈ wcel 1480 〈cop 3525 × cxp 4532 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-xp 4540 |
This theorem is referenced by: 3optocl 4612 ecopovsym 6518 ecopovsymg 6521 th3qlem2 6525 axaddcom 7671 |
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