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Theorem nffr 5117
Description: Bound-variable hypothesis builder for well-founded relations. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nffr.r 𝑥𝑅
nffr.a 𝑥𝐴
Assertion
Ref Expression
nffr 𝑥 𝑅 Fr 𝐴

Proof of Theorem nffr
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fr 5102 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏))
2 nfcv 2793 . . . . . 6 𝑥𝑎
3 nffr.a . . . . . 6 𝑥𝐴
42, 3nfss 3629 . . . . 5 𝑥 𝑎𝐴
5 nfv 1883 . . . . 5 𝑥 𝑎 ≠ ∅
64, 5nfan 1868 . . . 4 𝑥(𝑎𝐴𝑎 ≠ ∅)
7 nfcv 2793 . . . . . . . 8 𝑥𝑐
8 nffr.r . . . . . . . 8 𝑥𝑅
9 nfcv 2793 . . . . . . . 8 𝑥𝑏
107, 8, 9nfbr 4732 . . . . . . 7 𝑥 𝑐𝑅𝑏
1110nfn 1824 . . . . . 6 𝑥 ¬ 𝑐𝑅𝑏
122, 11nfral 2974 . . . . 5 𝑥𝑐𝑎 ¬ 𝑐𝑅𝑏
132, 12nfrex 3036 . . . 4 𝑥𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏
146, 13nfim 1865 . . 3 𝑥((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏)
1514nfal 2191 . 2 𝑥𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏)
161, 15nfxfr 1819 1 𝑥 𝑅 Fr 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1521  wnf 1748  wnfc 2780  wne 2823  wral 2941  wrex 2942  wss 3607  c0 3948   class class class wbr 4685   Fr wfr 5099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-fr 5102
This theorem is referenced by:  nfwe  5119
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