MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nffr Structured version   Visualization version   GIF version

Theorem nffr 5656
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 5637 . 2 (𝑅 Fr 𝐴 ↔ ∀𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏))
2 nfcv 2892 . . . . . 6 𝑥𝑎
3 nffr.a . . . . . 6 𝑥𝐴
42, 3nfss 3972 . . . . 5 𝑥 𝑎𝐴
5 nfv 1910 . . . . 5 𝑥 𝑎 ≠ ∅
64, 5nfan 1895 . . . 4 𝑥(𝑎𝐴𝑎 ≠ ∅)
7 nfcv 2892 . . . . . . . 8 𝑥𝑐
8 nffr.r . . . . . . . 8 𝑥𝑅
9 nfcv 2892 . . . . . . . 8 𝑥𝑏
107, 8, 9nfbr 5200 . . . . . . 7 𝑥 𝑐𝑅𝑏
1110nfn 1853 . . . . . 6 𝑥 ¬ 𝑐𝑅𝑏
122, 11nfralw 3299 . . . . 5 𝑥𝑐𝑎 ¬ 𝑐𝑅𝑏
132, 12nfrexw 3301 . . . 4 𝑥𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏
146, 13nfim 1892 . . 3 𝑥((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏)
1514nfal 2312 . 2 𝑥𝑎((𝑎𝐴𝑎 ≠ ∅) → ∃𝑏𝑎𝑐𝑎 ¬ 𝑐𝑅𝑏)
161, 15nfxfr 1848 1 𝑥 𝑅 Fr 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wal 1532  wnf 1778  wnfc 2876  wne 2930  wral 3051  wrex 3060  wss 3947  c0 4325   class class class wbr 5153   Fr wfr 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-fr 5637
This theorem is referenced by:  nfwe  5658
  Copyright terms: Public domain W3C validator