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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 6991 | . 2 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
5 | caovord2.3 | . . . 4 ⊢ 𝐶 ∈ V | |
6 | caovord2.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
7 | 5, 1, 6 | caovcom 6977 | . . 3 ⊢ (𝐶𝐹𝐴) = (𝐴𝐹𝐶) |
8 | 5, 2, 6 | caovcom 6977 | . . 3 ⊢ (𝐶𝐹𝐵) = (𝐵𝐹𝐶) |
9 | 7, 8 | breq12i 4793 | . 2 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)) |
10 | 4, 9 | syl6bb 276 | 1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1630 ∈ wcel 2144 Vcvv 3349 class class class wbr 4784 (class class class)co 6792 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ral 3065 df-rex 3066 df-rab 3069 df-v 3351 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-br 4785 df-iota 5994 df-fv 6039 df-ov 6795 |
This theorem is referenced by: caovord3 6993 genpnmax 10030 addclprlem1 10039 mulclprlem 10042 distrlem4pr 10049 ltexprlem6 10064 reclem3pr 10072 ltsosr 10116 |
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