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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 3060 | . . . . . . . . . 10 |
5 | ifeq1 3535 | . . . . . . . . . 10 | |
6 | 4, 5 | ax-mp 5 | . . . . . . . . 9 |
7 | 6 | mpteq2i 4085 | . . . . . . . 8 |
8 | seqeq3 10418 | . . . . . . . 8 | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 |
10 | 9 | breq1i 4005 | . . . . . 6 |
11 | 10 | 3anbi3i 1192 | . . . . 5 DECID DECID |
12 | 11 | rexbii 2482 | . . . 4 DECID DECID |
13 | 1, 2, 3 | cbvcsb 3060 | . . . . . . . . . . . 12 |
14 | ifeq1 3535 | . . . . . . . . . . . 12 | |
15 | 13, 14 | ax-mp 5 | . . . . . . . . . . 11 |
16 | 15 | mpteq2i 4085 | . . . . . . . . . 10 |
17 | seqeq3 10418 | . . . . . . . . . 10 | |
18 | 16, 17 | ax-mp 5 | . . . . . . . . 9 |
19 | 18 | fveq1i 5508 | . . . . . . . 8 |
20 | 19 | eqeq2i 2186 | . . . . . . 7 |
21 | 20 | anbi2i 457 | . . . . . 6 |
22 | 21 | exbii 1603 | . . . . 5 |
23 | 22 | rexbii 2482 | . . . 4 |
24 | 12, 23 | orbi12i 764 | . . 3 DECID DECID |
25 | 24 | iotabii 5192 | . 2 DECID DECID |
26 | df-sumdc 11328 | . 2 DECID | |
27 | df-sumdc 11328 | . 2 DECID | |
28 | 25, 26, 27 | 3eqtr4i 2206 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wo 708 DECID wdc 834 w3a 978 wceq 1353 wex 1490 wcel 2146 wnfc 2304 wral 2453 wrex 2454 csb 3055 wss 3127 cif 3532 class class class wbr 3998 cmpt 4059 cio 5168 wf1o 5207 cfv 5208 (class class class)co 5865 cc0 7786 c1 7787 caddc 7789 cle 7967 cn 8890 cz 9224 cuz 9499 cfz 9977 cseq 10413 cli 11252 csu 11327 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-ext 2157 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-un 3131 df-in 3133 df-ss 3140 df-if 3533 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-mpt 4061 df-cnv 4628 df-dm 4630 df-rn 4631 df-res 4632 df-iota 5170 df-fv 5216 df-ov 5868 df-oprab 5869 df-mpo 5870 df-recs 6296 df-frec 6382 df-seqfrec 10414 df-sumdc 11328 |
This theorem is referenced by: cbvsumv 11335 cbvsumi 11336 fsumsplitf 11382 |
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