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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 6743 | . 2 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
5 | caovord2.3 | . . . 4 ⊢ 𝐶 ∈ V | |
6 | caovord2.com | . . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
7 | 5, 1, 6 | caovcom 6729 | . . 3 ⊢ (𝐶𝐹𝐴) = (𝐴𝐹𝐶) |
8 | 5, 2, 6 | caovcom 6729 | . . 3 ⊢ (𝐶𝐹𝐵) = (𝐵𝐹𝐶) |
9 | 7, 8 | breq12i 4592 | . 2 ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶)) |
10 | 4, 9 | syl6bb 275 | 1 ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 (class class class)co 6549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: caovord3 6745 genpnmax 9708 addclprlem1 9717 mulclprlem 9720 distrlem4pr 9727 ltexprlem6 9742 reclem3pr 9750 ltsosr 9794 |
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