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| Mirrors > Home > MPE Home > Th. List > caovass | Structured version Visualization version 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 1544 | . . 3 ⊢ ⊤ | |
| 5 | caovass.4 | . . . . 5 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)) | |
| 6 | 5 | a1i 11 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) | 
| 7 | 6 | caovassg 7631 | . . 3 ⊢ ((⊤ ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) | 
| 8 | 4, 7 | mpan 690 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) | 
| 9 | 1, 2, 3, 8 | mp3an 1463 | 1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 Vcvv 3480 (class class class)co 7431 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: caov32 7660 caov12 7661 caov31 7662 caov13 7663 caov4 7664 caov411 7665 caovdilem 7668 caovmo 7670 | 
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