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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 230 | . . 3 ⊢ (〈𝑧, 𝑤〉 = 𝐵 → ((𝐴 ∈ 𝑅 → 𝜓) ↔ (𝐴 ∈ 𝑅 → 𝜒))) |
4 | 2optocl.2 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | 4 | imbi2d 230 | . . . . 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 115 | . . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → ((𝑧 ∈ 𝐶 ∧ 𝑤 ∈ 𝐷) → 𝜑)) |
8 | 1, 5, 7 | optocl 4698 | . . . 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 4698 | . 2 ⊢ (𝐵 ∈ 𝑅 → (𝐴 ∈ 𝑅 → 𝜒)) |
11 | 10 | impcom 125 | 1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑅) → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 〈cop 3594 × cxp 4620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-opab 4062 df-xp 4628 |
This theorem is referenced by: 3optocl 4700 ecopovsym 6624 ecopovsymg 6627 th3qlem2 6631 axaddcom 7847 |
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