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Theorem dfrnf 5246
 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:   ,

Proof of Theorem dfrnf
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrn2 5196 . 2
2 nfcv 2619 . . . . 5
3 dfrnf.1 . . . . 5
4 nfcv 2619 . . . . 5
52, 3, 4nfbr 4496 . . . 4
6 nfv 1707 . . . 4
7 breq1 4455 . . . 4
85, 6, 7cbvex 2022 . . 3
98abbii 2591 . 2
10 nfcv 2619 . . . . 5
11 dfrnf.2 . . . . 5
12 nfcv 2619 . . . . 5
1310, 11, 12nfbr 4496 . . . 4
1413nfex 1948 . . 3
15 nfv 1707 . . 3
16 breq2 4456 . . . 4
1716exbidv 1714 . . 3
1814, 15, 17cbvab 2598 . 2
191, 9, 183eqtri 2490 1
 Colors of variables: wff setvar class Syntax hints:  =wceq 1395  E.wex 1612  {cab 2442  F/_wnfc 2605   class class class wbr 4452  rancrn 5005 This theorem is referenced by:  rnopab  5252 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-cnv 5012  df-dm 5014  df-rn 5015
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