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Mirrors > Home > ILE Home > Th. List > cbvmpt | Unicode version |
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by NM, 11-Sep-2011.) |
Ref | Expression |
---|---|
cbvmpt.1 | |
cbvmpt.2 | |
cbvmpt.3 |
Ref | Expression |
---|---|
cbvmpt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1493 | . . . 4 | |
2 | nfv 1493 | . . . . 5 | |
3 | nfs1v 1892 | . . . . 5 | |
4 | 2, 3 | nfan 1529 | . . . 4 |
5 | eleq1 2180 | . . . . 5 | |
6 | sbequ12 1729 | . . . . 5 | |
7 | 5, 6 | anbi12d 464 | . . . 4 |
8 | 1, 4, 7 | cbvopab1 3971 | . . 3 |
9 | nfv 1493 | . . . . 5 | |
10 | cbvmpt.1 | . . . . . . 7 | |
11 | 10 | nfeq2 2270 | . . . . . 6 |
12 | 11 | nfsb 1899 | . . . . 5 |
13 | 9, 12 | nfan 1529 | . . . 4 |
14 | nfv 1493 | . . . 4 | |
15 | eleq1 2180 | . . . . 5 | |
16 | sbequ 1796 | . . . . . 6 | |
17 | cbvmpt.2 | . . . . . . . 8 | |
18 | 17 | nfeq2 2270 | . . . . . . 7 |
19 | cbvmpt.3 | . . . . . . . 8 | |
20 | 19 | eqeq2d 2129 | . . . . . . 7 |
21 | 18, 20 | sbie 1749 | . . . . . 6 |
22 | 16, 21 | syl6bb 195 | . . . . 5 |
23 | 15, 22 | anbi12d 464 | . . . 4 |
24 | 13, 14, 23 | cbvopab1 3971 | . . 3 |
25 | 8, 24 | eqtri 2138 | . 2 |
26 | df-mpt 3961 | . 2 | |
27 | df-mpt 3961 | . 2 | |
28 | 25, 26, 27 | 3eqtr4i 2148 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wsb 1720 wnfc 2245 copab 3958 cmpt 3959 |
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-v 2662 df-un 3045 df-sn 3503 df-pr 3504 df-op 3506 df-opab 3960 df-mpt 3961 |
This theorem is referenced by: cbvmptv 3994 dffn5imf 5444 fvmpts 5467 fvmpt2 5472 mptfvex 5474 fmptcof 5555 fmptcos 5556 fliftfuns 5667 offval2 5965 qliftfuns 6481 summodclem2a 11118 zsumdc 11121 fsum3cvg2 11131 cnmpt1t 12381 fsumcncntop 12652 limcmpted 12728 |
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