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Theorem dfrnf 4623
 Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dfrnf.1
dfrnf.2
Assertion
Ref Expression
dfrnf
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem dfrnf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrn2 4571 . 2
2 nfcv 2223 . . . . 5
3 dfrnf.1 . . . . 5
4 nfcv 2223 . . . . 5
52, 3, 4nfbr 3849 . . . 4
6 nfv 1462 . . . 4
7 breq1 3808 . . . 4
85, 6, 7cbvex 1681 . . 3
98abbii 2198 . 2
10 nfcv 2223 . . . . 5
11 dfrnf.2 . . . . 5
12 nfcv 2223 . . . . 5
1310, 11, 12nfbr 3849 . . . 4
1413nfex 1569 . . 3
15 nfv 1462 . . 3
16 breq2 3809 . . . 4
1716exbidv 1748 . . 3
1814, 15, 17cbvab 2205 . 2
191, 9, 183eqtri 2107 1
 Colors of variables: wff set class Syntax hints:   wceq 1285  wex 1422  cab 2069  wnfc 2210   class class class wbr 3805   crn 4392 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-cnv 4399  df-dm 4401  df-rn 4402 This theorem is referenced by:  rnopab  4629
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