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Theorem cbvmptf 4493
 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 Thierry Arnoux, 9-Mar-2017.)
Hypotheses
Ref Expression
cbvmptf.1
cbvmptf.2
cbvmptf.3
cbvmptf.4
cbvmptf.5
Assertion
Ref Expression
cbvmptf
Distinct variable group:   ,
Allowed substitution hints:   (,)   (,)   (,)

Proof of Theorem cbvmptf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1761 . . . 4
2 cbvmptf.1 . . . . . 6
32nfcri 2586 . . . . 5
4 nfs1v 2266 . . . . 5
53, 4nfan 2011 . . . 4
6 eleq1 2517 . . . . 5
7 sbequ12 2083 . . . . 5
86, 7anbi12d 717 . . . 4
91, 5, 8cbvopab1 4473 . . 3
10 cbvmptf.2 . . . . . 6
1110nfcri 2586 . . . . 5
12 cbvmptf.3 . . . . . . 7
1312nfeq2 2607 . . . . . 6
1413nfsb 2269 . . . . 5
1511, 14nfan 2011 . . . 4
16 nfv 1761 . . . 4
17 eleq1 2517 . . . . 5
18 cbvmptf.4 . . . . . . 7
1918nfeq2 2607 . . . . . 6
20 cbvmptf.5 . . . . . . 7
2120eqeq2d 2461 . . . . . 6
2219, 21sbhypf 3095 . . . . 5
2317, 22anbi12d 717 . . . 4
2415, 16, 23cbvopab1 4473 . . 3
259, 24eqtri 2473 . 2
26 df-mpt 4463 . 2
27 df-mpt 4463 . 2
2825, 26, 273eqtr4i 2483 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371   wceq 1444  wsb 1797   wcel 1887  wnfc 2579  copab 4460   cmpt 4461 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-opab 4462  df-mpt 4463 This theorem is referenced by:  fvmpt2f  5949  offval2f  6543  suppss2f  28238  fmptdF  28255  resmptf  28257  acunirnmpt2f  28263  funcnv4mpt  28273  cbvesum  28863  esumpfinvalf  28897  binomcxplemdvbinom  36702  binomcxplemdvsum  36704  binomcxplemnotnn0  36705  sge0iunmptlemre  38257
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