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Theorem cbvmpt2x 6356
 Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6357 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 1728 . . . . 5
2 cbvmpt2x.1 . . . . . 6
32nfcri 2557 . . . . 5
41, 3nfan 1956 . . . 4
5 cbvmpt2x.3 . . . . 5
65nfeq2 2581 . . . 4
74, 6nfan 1956 . . 3
8 nfv 1728 . . . . 5
9 nfcv 2564 . . . . . 6
109nfcri 2557 . . . . 5
118, 10nfan 1956 . . . 4
12 cbvmpt2x.4 . . . . 5
1312nfeq2 2581 . . . 4
1411, 13nfan 1956 . . 3
15 nfv 1728 . . . . 5
16 cbvmpt2x.2 . . . . . 6
1716nfcri 2557 . . . . 5
1815, 17nfan 1956 . . . 4
19 cbvmpt2x.5 . . . . 5
2019nfeq2 2581 . . . 4
2118, 20nfan 1956 . . 3
22 nfv 1728 . . . 4
23 cbvmpt2x.6 . . . . 5
2423nfeq2 2581 . . . 4
2522, 24nfan 1956 . . 3
26 eleq1 2474 . . . . . 6
2726adantr 463 . . . . 5
28 cbvmpt2x.7 . . . . . . 7
2928eleq2d 2472 . . . . . 6
30 eleq1 2474 . . . . . 6
3129, 30sylan9bb 698 . . . . 5
3227, 31anbi12d 709 . . . 4
33 cbvmpt2x.8 . . . . 5
3433eqeq2d 2416 . . . 4
3532, 34anbi12d 709 . . 3
367, 14, 21, 25, 35cbvoprab12 6352 . 2
37 df-mpt2 6283 . 2
38 df-mpt2 6283 . 2
3936, 37, 383eqtr4i 2441 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   wceq 1405   wcel 1842  wnfc 2550  coprab 6279   cmpt2 6280 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-opab 4454  df-oprab 6282  df-mpt2 6283 This theorem is referenced by:  cbvmpt2  6357  mpt2mptsx  6847  dmmpt2ssx  6849  gsumcom2  17324  ptcmpg  20849
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