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Theorem cbvmpox 5815
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 5816 allows to be a function of . (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
cbvmpox.1
cbvmpox.2
cbvmpox.3
cbvmpox.4
cbvmpox.5
cbvmpox.6
cbvmpox.7
cbvmpox.8
Assertion
Ref Expression
cbvmpox
Distinct variable groups:   ,,,,   ,   ,
Allowed substitution hints:   (,,)   (,,,)   (,,)   (,,,)

Proof of Theorem cbvmpox
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1491 . . . . 5
2 cbvmpox.1 . . . . . 6
32nfcri 2250 . . . . 5
41, 3nfan 1527 . . . 4
5 cbvmpox.3 . . . . 5
65nfeq2 2268 . . . 4
74, 6nfan 1527 . . 3
8 nfv 1491 . . . . 5
9 nfcv 2256 . . . . . 6
109nfcri 2250 . . . . 5
118, 10nfan 1527 . . . 4
12 cbvmpox.4 . . . . 5
1312nfeq2 2268 . . . 4
1411, 13nfan 1527 . . 3
15 nfv 1491 . . . . 5
16 cbvmpox.2 . . . . . 6
1716nfcri 2250 . . . . 5
1815, 17nfan 1527 . . . 4
19 cbvmpox.5 . . . . 5
2019nfeq2 2268 . . . 4
2118, 20nfan 1527 . . 3
22 nfv 1491 . . . 4
23 cbvmpox.6 . . . . 5
2423nfeq2 2268 . . . 4
2522, 24nfan 1527 . . 3
26 eleq1 2178 . . . . . 6
2726adantr 272 . . . . 5
28 cbvmpox.7 . . . . . . 7
2928eleq2d 2185 . . . . . 6
30 eleq1 2178 . . . . . 6
3129, 30sylan9bb 455 . . . . 5
3227, 31anbi12d 462 . . . 4
33 cbvmpox.8 . . . . 5
3433eqeq2d 2127 . . . 4
3532, 34anbi12d 462 . . 3
367, 14, 21, 25, 35cbvoprab12 5811 . 2
37 df-mpo 5745 . 2
38 df-mpo 5745 . 2
3936, 37, 383eqtr4i 2146 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1314   wcel 1463  wnfc 2243  coprab 5741   cmpo 5742 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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-oprab 5744  df-mpo 5745 This theorem is referenced by:  cbvmpo  5816  mpomptsx  6061  dmmpossx  6063
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