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Mirrors > Home > ILE Home > Th. List > add12d | GIF version |
Description: Commutative/associative law that swaps the first two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
addd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addd.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
add12d | ⊢ (𝜑 → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | addd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | addd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | add12 8047 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))) | |
5 | 1, 2, 3, 4 | syl3anc 1227 | 1 ⊢ (𝜑 → (𝐴 + (𝐵 + 𝐶)) = (𝐵 + (𝐴 + 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 (class class class)co 5836 ℂcc 7742 + caddc 7747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-addcom 7844 ax-addass 7846 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-rex 2448 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-iota 5147 df-fv 5190 df-ov 5839 |
This theorem is referenced by: subsub2 8117 bdtrilem 11166 |
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