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Theorem cbvmpt 3879
 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.)
Hypotheses
Ref Expression
cbvmpt.1
cbvmpt.2
cbvmpt.3
Assertion
Ref Expression
cbvmpt
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)

Proof of Theorem cbvmpt
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1437 . . . 4
2 nfv 1437 . . . . 5
3 nfs1v 1831 . . . . 5
42, 3nfan 1473 . . . 4
5 eleq1 2116 . . . . 5
6 sbequ12 1670 . . . . 5
75, 6anbi12d 450 . . . 4
81, 4, 7cbvopab1 3858 . . 3
9 nfv 1437 . . . . 5
10 cbvmpt.1 . . . . . . 7
1110nfeq2 2205 . . . . . 6
1211nfsb 1838 . . . . 5
139, 12nfan 1473 . . . 4
14 nfv 1437 . . . 4
15 eleq1 2116 . . . . 5
16 sbequ 1737 . . . . . 6
17 cbvmpt.2 . . . . . . . 8
1817nfeq2 2205 . . . . . . 7
19 cbvmpt.3 . . . . . . . 8
2019eqeq2d 2067 . . . . . . 7
2118, 20sbie 1690 . . . . . 6
2216, 21syl6bb 189 . . . . 5
2315, 22anbi12d 450 . . . 4
2413, 14, 23cbvopab1 3858 . . 3
258, 24eqtri 2076 . 2
26 df-mpt 3848 . 2
27 df-mpt 3848 . 2
2825, 26, 273eqtr4i 2086 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wceq 1259   wcel 1409  wsb 1661  wnfc 2181  copab 3845   cmpt 3846 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-mpt 3848 This theorem is referenced by:  cbvmptv  3880  dffn5imf  5256  fvmpts  5278  fvmpt2  5282  mptfvex  5284  fmptcof  5359  fmptcos  5360  fliftfuns  5466  offval2  5754  qliftfuns  6221
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