![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > caovord2 | Structured version Visualization version GIF version |
Description: Operation ordering law with commuted arguments. (Contributed by NM, 27-Feb-1996.) |
Ref | Expression |
---|---|
caovord.1 | ⊢ 𝐴 ∈ V |
caovord.2 | ⊢ 𝐵 ∈ V |
caovord.3 | ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) |
caovord2.3 | ⊢ 𝐶 ∈ V |
caovord2.com | ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) |
Ref | Expression |
---|---|
caovord2 | ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovord.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | caovord.2 | . . 3 ⊢ 𝐵 ∈ V | |
3 | caovord.3 | . . 3 ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) | |
4 | 1, 2, 3 | caovord 7611 | . 2 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
5 | caovord2.3 | . . . 4 ⊢ 𝐶 ∈ V | |
6 | caovord2.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
7 | 5, 1, 6 | caovcom 7597 | . . 3 ⊢ (𝐶𝐹𝐴) = (𝐴𝐹𝐶) |
8 | 5, 2, 6 | caovcom 7597 | . . 3 ⊢ (𝐶𝐹𝐵) = (𝐵𝐹𝐶) |
9 | 7, 8 | breq12i 5147 | . 2 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)) |
10 | 4, 9 | bitrdi 287 | 1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 Vcvv 3466 class class class wbr 5138 (class class class)co 7401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-iota 6485 df-fv 6541 df-ov 7404 |
This theorem is referenced by: caovord3 7613 genpnmax 10997 addclprlem1 11006 mulclprlem 11009 distrlem4pr 11016 ltexprlem6 11031 reclem3pr 11039 ltsosr 11084 |
Copyright terms: Public domain | W3C validator |