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Mirrors > Home > ILE Home > Th. List > caovcang | GIF version |
Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovcang.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) |
Ref | Expression |
---|---|
caovcang | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovcang.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) | |
2 | 1 | ralrimivvva 2444 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) |
3 | oveq1 5539 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦)) | |
4 | oveq1 5539 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑧) = (𝐴𝐹𝑧)) | |
5 | 3, 4 | eqeq12d 2095 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ (𝐴𝐹𝑦) = (𝐴𝐹𝑧))) |
6 | 5 | bibi1d 231 | . . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧) ↔ ((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ 𝑦 = 𝑧))) |
7 | oveq2 5540 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵)) | |
8 | 7 | eqeq1d 2089 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝑧))) |
9 | eqeq1 2087 | . . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝑧 ↔ 𝐵 = 𝑧)) | |
10 | 8, 9 | bibi12d 233 | . . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ 𝐵 = 𝑧))) |
11 | oveq2 5540 | . . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝐹𝑧) = (𝐴𝐹𝐶)) | |
12 | 11 | eqeq2d 2092 | . . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝐶))) |
13 | eqeq2 2090 | . . . 4 ⊢ (𝑧 = 𝐶 → (𝐵 = 𝑧 ↔ 𝐵 = 𝐶)) | |
14 | 12, 13 | bibi12d 233 | . . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ 𝐵 = 𝑧) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))) |
15 | 6, 10, 14 | rspc3v 2716 | . 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (∀𝑥 ∈ 𝑇 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))) |
16 | 2, 15 | mpan9 275 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 919 = wceq 1284 ∈ wcel 1433 ∀wral 2348 (class class class)co 5532 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: caovcand 5683 |
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