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Mirrors > Home > ILE Home > Th. List > caovordid | GIF version |
Description: Convert an operation ordering law to class notation. (Contributed by Mario Carneiro, 31-Dec-2014.) |
Ref | Expression |
---|---|
caovordig.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) |
caovordid.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caovordid.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
caovordid.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
Ref | Expression |
---|---|
caovordid | ⊢ (𝜑 → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 ⊢ (𝜑 → 𝜑) | |
2 | caovordid.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
3 | caovordid.3 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
4 | caovordid.4 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
5 | caovordig.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 → (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) | |
6 | 5 | caovordig 5890 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
7 | 1, 2, 3, 4, 6 | syl13anc 1201 | 1 ⊢ (𝜑 → (𝐴𝑅𝐵 → (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 945 ∈ wcel 1463 class class class wbr 3895 (class class class)co 5728 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-iota 5046 df-fv 5089 df-ov 5731 |
This theorem is referenced by: (None) |
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