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 7515 | . 2 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
5 | caovord2.3 | . . . 4 ⊢ 𝐶 ∈ V | |
6 | caovord2.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
7 | 5, 1, 6 | caovcom 7501 | . . 3 ⊢ (𝐶𝐹𝐴) = (𝐴𝐹𝐶) |
8 | 5, 2, 6 | caovcom 7501 | . . 3 ⊢ (𝐶𝐹𝐵) = (𝐵𝐹𝐶) |
9 | 7, 8 | breq12i 5090 | . 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 1539 ∈ wcel 2104 Vcvv 3437 class class class wbr 5081 (class class class)co 7307 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-iota 6410 df-fv 6466 df-ov 7310 |
This theorem is referenced by: caovord3 7517 genpnmax 10809 addclprlem1 10818 mulclprlem 10821 distrlem4pr 10828 ltexprlem6 10843 reclem3pr 10851 ltsosr 10896 |
Copyright terms: Public domain | W3C validator |