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Mirrors > Home > MPE Home > Th. List > caovord2d | Structured version Visualization version GIF version |
Description: Operation ordering law with commuted arguments. (Contributed by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovordg.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) |
caovordd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caovordd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
caovordd.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
caovord2d.com | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) |
Ref | Expression |
---|---|
caovord2d | ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovordg.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) | |
2 | caovordd.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | caovordd.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
4 | caovordd.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
5 | 1, 2, 3, 4 | caovordd 6842 | . 2 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
6 | caovord2d.com | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | |
7 | 6, 4, 2 | caovcomd 6830 | . . 3 ⊢ (𝜑 → (𝐶𝐹𝐴) = (𝐴𝐹𝐶)) |
8 | 6, 4, 3 | caovcomd 6830 | . . 3 ⊢ (𝜑 → (𝐶𝐹𝐵) = (𝐵𝐹𝐶)) |
9 | 7, 8 | breq12d 4666 | . 2 ⊢ (𝜑 → ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
10 | 5, 9 | bitrd 268 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐶)𝑅(𝐵𝐹𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: caovord3d 6844 |
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