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Mirrors > Home > ILE Home > Th. List > add4i | GIF version |
Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 9-May-1999.) |
Ref | Expression |
---|---|
add.1 | ⊢ 𝐴 ∈ ℂ |
add.2 | ⊢ 𝐵 ∈ ℂ |
add.3 | ⊢ 𝐶 ∈ ℂ |
add4.4 | ⊢ 𝐷 ∈ ℂ |
Ref | Expression |
---|---|
add4i | ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | add.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | add.2 | . 2 ⊢ 𝐵 ∈ ℂ | |
3 | add.3 | . 2 ⊢ 𝐶 ∈ ℂ | |
4 | add4.4 | . 2 ⊢ 𝐷 ∈ ℂ | |
5 | add4 8108 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐶 ∈ ℂ ∧ 𝐷 ∈ ℂ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))) | |
6 | 1, 2, 3, 4, 5 | mp4an 427 | 1 ⊢ ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∈ wcel 2148 (class class class)co 5869 ℂcc 7800 + caddc 7805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-addcl 7898 ax-addcom 7902 ax-addass 7904 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-iota 5174 df-fv 5220 df-ov 5872 |
This theorem is referenced by: add42i 8113 negdii 8231 numma 9416 |
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