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Mirrors > Home > ILE Home > Th. List > cbvsum | Unicode version |
Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.) |
Ref | Expression |
---|---|
cbvsum.1 | |
cbvsum.2 | |
cbvsum.3 | |
cbvsum.4 | |
cbvsum.5 |
Ref | Expression |
---|---|
cbvsum |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvsum.4 | . . . . . . . . . . 11 | |
2 | cbvsum.5 | . . . . . . . . . . 11 | |
3 | cbvsum.1 | . . . . . . . . . . 11 | |
4 | 1, 2, 3 | cbvcsb 2979 | . . . . . . . . . 10 |
5 | ifeq1 3447 | . . . . . . . . . 10 | |
6 | 4, 5 | ax-mp 5 | . . . . . . . . 9 |
7 | 6 | mpteq2i 3985 | . . . . . . . 8 |
8 | seqeq3 10191 | . . . . . . . 8 | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 |
10 | 9 | breq1i 3906 | . . . . . 6 |
11 | 10 | 3anbi3i 1159 | . . . . 5 DECID DECID |
12 | 11 | rexbii 2419 | . . . 4 DECID DECID |
13 | 1, 2, 3 | cbvcsb 2979 | . . . . . . . . . . . 12 |
14 | ifeq1 3447 | . . . . . . . . . . . 12 | |
15 | 13, 14 | ax-mp 5 | . . . . . . . . . . 11 |
16 | 15 | mpteq2i 3985 | . . . . . . . . . 10 |
17 | seqeq3 10191 | . . . . . . . . . 10 | |
18 | 16, 17 | ax-mp 5 | . . . . . . . . 9 |
19 | 18 | fveq1i 5390 | . . . . . . . 8 |
20 | 19 | eqeq2i 2128 | . . . . . . 7 |
21 | 20 | anbi2i 452 | . . . . . 6 |
22 | 21 | exbii 1569 | . . . . 5 |
23 | 22 | rexbii 2419 | . . . 4 |
24 | 12, 23 | orbi12i 738 | . . 3 DECID DECID |
25 | 24 | iotabii 5080 | . 2 DECID DECID |
26 | df-sumdc 11091 | . 2 DECID | |
27 | df-sumdc 11091 | . 2 DECID | |
28 | 25, 26, 27 | 3eqtr4i 2148 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 DECID wdc 804 w3a 947 wceq 1316 wex 1453 wcel 1465 wnfc 2245 wral 2393 wrex 2394 csb 2975 wss 3041 cif 3444 class class class wbr 3899 cmpt 3959 cio 5056 wf1o 5092 cfv 5093 (class class class)co 5742 cc0 7588 c1 7589 caddc 7591 cle 7769 cn 8688 cz 9022 cuz 9294 cfz 9758 cseq 10186 cli 11015 csu 11090 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-if 3445 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-iota 5058 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-recs 6170 df-frec 6256 df-seqfrec 10187 df-sumdc 11091 |
This theorem is referenced by: cbvsumv 11098 cbvsumi 11099 fsumsplitf 11145 |
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