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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 7609 | . . 3 ⊢ ((⊤ ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
8 | 4, 7 | mpan 687 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
9 | 1, 2, 3, 8 | mp3an 1460 | 1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1540 ⊤wtru 1541 ∈ wcel 2105 Vcvv 3473 (class class class)co 7412 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7415 |
This theorem is referenced by: caov32 7638 caov12 7639 caov31 7640 caov13 7641 caov4 7642 caov411 7643 caovdilem 7646 caovmo 7648 |
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