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Theorem dfrnf 5110
 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 5061 . 2
2 nfcv 2574 . . . . 5
3 dfrnf.1 . . . . 5
4 nfcv 2574 . . . . 5
52, 3, 4nfbr 4258 . . . 4
6 nfv 1630 . . . 4
7 breq1 4217 . . . 4
85, 6, 7cbvex 1984 . . 3
98abbii 2550 . 2
10 nfcv 2574 . . . . 5
11 dfrnf.2 . . . . 5
12 nfcv 2574 . . . . 5
1310, 11, 12nfbr 4258 . . . 4
1413nfex 1866 . . 3
15 nfv 1630 . . 3
16 breq2 4218 . . . 4
1716exbidv 1637 . . 3
1814, 15, 17cbvab 2556 . 2
191, 9, 183eqtri 2462 1
 Colors of variables: wff set class Syntax hints:  wex 1551   wceq 1653  cab 2424  wnfc 2561   class class class wbr 4214   crn 4881 This theorem is referenced by:  rnopab  5117 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-cnv 4888  df-dm 4890  df-rn 4891
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