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Theorem nffr 5377
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 5362 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏))
2 nfcv 2926 . . . . . 6 𝑥𝑎
3 nffr.a . . . . . 6 𝑥𝐴
42, 3nfss 3845 . . . . 5 𝑥 𝑎𝐴
5 nfv 1873 . . . . 5 𝑥 𝑎 ≠ ∅
64, 5nfan 1862 . . . 4 𝑥(𝑎𝐴𝑎 ≠ ∅)
7 nfcv 2926 . . . . . . . 8 𝑥𝑐
8 nffr.r . . . . . . . 8 𝑥𝑅
9 nfcv 2926 . . . . . . . 8 𝑥𝑏
107, 8, 9nfbr 4972 . . . . . . 7 𝑥 𝑐𝑅𝑏
1110nfn 1819 . . . . . 6 𝑥 ¬ 𝑐𝑅𝑏
122, 11nfral 3168 . . . . 5 𝑥𝑐𝑎 ¬ 𝑐𝑅𝑏
132, 12nfrex 3247 . . . 4 𝑥𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏
146, 13nfim 1859 . . 3 𝑥((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏)
1514nfal 2263 . 2 𝑥𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏)
161, 15nfxfr 1815 1 𝑥 𝑅 Fr 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  wal 1505  wnf 1746  wnfc 2910  wne 2961  wral 3082  wrex 3083  wss 3823  c0 4172   class class class wbr 4925   Fr wfr 5359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-fr 5362
This theorem is referenced by:  nfwe  5379
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