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Mirrors > Home > ILE Home > Th. List > caovass | GIF version |
Description: Convert an operation associative law to class notation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 26-May-2014.) |
Ref | Expression |
---|---|
caovass.1 | ⊢ 𝐴 ∈ V |
caovass.2 | ⊢ 𝐵 ∈ V |
caovass.3 | ⊢ 𝐶 ∈ V |
caovass.4 | ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) |
Ref | Expression |
---|---|
caovass | ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovass.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | caovass.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | caovass.3 | . 2 ⊢ 𝐶 ∈ V | |
4 | tru 1335 | . . 3 ⊢ ⊤ | |
5 | caovass.4 | . . . . 5 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) | |
6 | 5 | a1i 9 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) |
7 | 6 | caovassg 5922 | . . 3 ⊢ ((⊤ ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
8 | 4, 7 | mpan 420 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
9 | 1, 2, 3, 8 | mp3an 1315 | 1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∧ w3a 962 = wceq 1331 ⊤wtru 1332 ∈ wcel 1480 Vcvv 2681 (class class class)co 5767 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
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-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: caov32 5951 caov12 5952 caov31 5953 caov13 5954 |
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