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Mirrors > Home > ILE Home > Th. List > caovdi | GIF version |
Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.) |
Ref | Expression |
---|---|
caovdi.1 | ⊢ 𝐴 ∈ V |
caovdi.2 | ⊢ 𝐵 ∈ V |
caovdi.3 | ⊢ 𝐶 ∈ V |
caovdi.4 | ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)) |
Ref | Expression |
---|---|
caovdi | ⊢ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovdi.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | caovdi.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | caovdi.3 | . 2 ⊢ 𝐶 ∈ V | |
4 | tru 1357 | . . 3 ⊢ ⊤ | |
5 | caovdi.4 | . . . . 5 ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)) | |
6 | 5 | a1i 9 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))) |
7 | 6 | caovdig 6048 | . . 3 ⊢ ((⊤ ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))) |
8 | 4, 7 | mpan 424 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))) |
9 | 1, 2, 3, 8 | mp3an 1337 | 1 ⊢ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∧ w3a 978 = wceq 1353 ⊤wtru 1354 ∈ wcel 2148 Vcvv 2737 (class class class)co 5874 |
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-ext 2159 |
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-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-iota 5178 df-fv 5224 df-ov 5877 |
This theorem is referenced by: (None) |
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