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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 1538 | . . 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 7619 | . . 3 ⊢ ((⊤ ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
8 | 4, 7 | mpan 689 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
9 | 1, 2, 3, 8 | mp3an 1458 | 1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∧ w3a 1085 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 Vcvv 3471 (class class class)co 7420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-ov 7423 |
This theorem is referenced by: caov32 7648 caov12 7649 caov31 7650 caov13 7651 caov4 7652 caov411 7653 caovdilem 7656 caovmo 7658 |
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