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Theorem cbvmpt2x 5582
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5583 allows to be a function of . (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
cbvmpt2x.1
cbvmpt2x.2
cbvmpt2x.3
cbvmpt2x.4
cbvmpt2x.5
cbvmpt2x.6
cbvmpt2x.7
cbvmpt2x.8
Assertion
Ref Expression
cbvmpt2x
Distinct variable groups:   ,,,,   ,   ,
Allowed substitution hints:   (,,)   (,,,)   (,,)   (,,,)

Proof of Theorem cbvmpt2x
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1421 . . . . 5
2 cbvmpt2x.1 . . . . . 6
32nfcri 2172 . . . . 5
41, 3nfan 1457 . . . 4
5 cbvmpt2x.3 . . . . 5
65nfeq2 2189 . . . 4
74, 6nfan 1457 . . 3
8 nfv 1421 . . . . 5
9 nfcv 2178 . . . . . 6
109nfcri 2172 . . . . 5
118, 10nfan 1457 . . . 4
12 cbvmpt2x.4 . . . . 5
1312nfeq2 2189 . . . 4
1411, 13nfan 1457 . . 3
15 nfv 1421 . . . . 5
16 cbvmpt2x.2 . . . . . 6
1716nfcri 2172 . . . . 5
1815, 17nfan 1457 . . . 4
19 cbvmpt2x.5 . . . . 5
2019nfeq2 2189 . . . 4
2118, 20nfan 1457 . . 3
22 nfv 1421 . . . 4
23 cbvmpt2x.6 . . . . 5
2423nfeq2 2189 . . . 4
2522, 24nfan 1457 . . 3
26 eleq1 2100 . . . . . 6
2726adantr 261 . . . . 5
28 cbvmpt2x.7 . . . . . . 7
2928eleq2d 2107 . . . . . 6
30 eleq1 2100 . . . . . 6
3129, 30sylan9bb 435 . . . . 5
3227, 31anbi12d 442 . . . 4
33 cbvmpt2x.8 . . . . 5
3433eqeq2d 2051 . . . 4
3532, 34anbi12d 442 . . 3
367, 14, 21, 25, 35cbvoprab12 5578 . 2
37 df-mpt2 5517 . 2
38 df-mpt2 5517 . 2
3936, 37, 383eqtr4i 2070 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243   wcel 1393  wnfc 2165  coprab 5513   cmpt2 5514 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-oprab 5516  df-mpt2 5517 This theorem is referenced by:  cbvmpt2  5583  mpt2mptsx  5823  dmmpt2ssx  5825
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