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Mirrors > Home > ILE Home > Th. List > add4 | Unicode version |
Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
add4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | add12 7913 | . . . . 5 | |
2 | 1 | 3expb 1182 | . . . 4 |
3 | 2 | oveq2d 5783 | . . 3 |
4 | 3 | adantll 467 | . 2 |
5 | addcl 7738 | . . 3 | |
6 | addass 7743 | . . . 4 | |
7 | 6 | 3expa 1181 | . . 3 |
8 | 5, 7 | sylan2 284 | . 2 |
9 | addcl 7738 | . . . 4 | |
10 | addass 7743 | . . . . 5 | |
11 | 10 | 3expa 1181 | . . . 4 |
12 | 9, 11 | sylan2 284 | . . 3 |
13 | 12 | an4s 577 | . 2 |
14 | 4, 8, 13 | 3eqtr4d 2180 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 (class class class)co 5767 cc 7611 caddc 7616 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-addcl 7709 ax-addcom 7713 ax-addass 7715 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: add42 7917 add4i 7920 add4d 7924 3dvds2dec 11552 opoe 11581 ptolemy 12894 |
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