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Mirrors > Home > ILE Home > Th. List > caovcand | GIF version |
Description: Convert an operation cancellation law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovcang.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) |
caovcand.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑇) |
caovcand.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
caovcand.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
Ref | Expression |
---|---|
caovcand | ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 ⊢ (𝜑 → 𝜑) | |
2 | caovcand.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑇) | |
3 | caovcand.3 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
4 | caovcand.4 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
5 | caovcang.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧)) | |
6 | 5 | caovcang 5864 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)) |
7 | 1, 2, 3, 4, 6 | syl13anc 1186 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 930 = wceq 1299 ∈ wcel 1448 (class class class)co 5706 |
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 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-un 3025 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-iota 5024 df-fv 5067 df-ov 5709 |
This theorem is referenced by: caovcanrd 5866 ecopovtrn 6456 ecopovtrng 6459 |
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