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Theorem dfrnf 5062
 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 5012 . 2
2 nfcv 2564 . . . . 5
3 dfrnf.1 . . . . 5
4 nfcv 2564 . . . . 5
52, 3, 4nfbr 4439 . . . 4
6 nfv 1728 . . . 4
7 breq1 4398 . . . 4
85, 6, 7cbvex 2049 . . 3
98abbii 2536 . 2
10 nfcv 2564 . . . . 5
11 dfrnf.2 . . . . 5
12 nfcv 2564 . . . . 5
1310, 11, 12nfbr 4439 . . . 4
1413nfex 1976 . . 3
15 nfv 1728 . . 3
16 breq2 4399 . . . 4
1716exbidv 1735 . . 3
1814, 15, 17cbvab 2543 . 2
191, 9, 183eqtri 2435 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1405  wex 1633  cab 2387  wnfc 2550   class class class wbr 4395   crn 4824 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-eu 2242  df-mo 2243  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-br 4396  df-opab 4454  df-cnv 4831  df-dm 4833  df-rn 4834 This theorem is referenced by:  rnopab  5068
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