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

Theorem nffr 4823
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  |-  F/_ x R
nffr.a  |-  F/_ x A
Assertion
Ref Expression
nffr  |-  F/ 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 4808 . 2  |-  ( R  Fr  A  <->  A. a
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b ) )
2 nfcv 2584 . . . . . 6  |-  F/_ x
a
3 nffr.a . . . . . 6  |-  F/_ x A
42, 3nfss 3457 . . . . 5  |-  F/ x  a  C_  A
5 nfv 1751 . . . . 5  |-  F/ x  a  =/=  (/)
64, 5nfan 1984 . . . 4  |-  F/ x
( a  C_  A  /\  a  =/=  (/) )
7 nfcv 2584 . . . . . . . 8  |-  F/_ x
c
8 nffr.r . . . . . . . 8  |-  F/_ x R
9 nfcv 2584 . . . . . . . 8  |-  F/_ x
b
107, 8, 9nfbr 4465 . . . . . . 7  |-  F/ x  c R b
1110nfn 1956 . . . . . 6  |-  F/ x  -.  c R b
122, 11nfral 2811 . . . . 5  |-  F/ x A. c  e.  a  -.  c R b
132, 12nfrex 2888 . . . 4  |-  F/ x E. b  e.  a  A. c  e.  a  -.  c R b
146, 13nfim 1976 . . 3  |-  F/ x
( ( a  C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b )
1514nfal 2003 . 2  |-  F/ x A. a ( ( a 
C_  A  /\  a  =/=  (/) )  ->  E. b  e.  a  A. c  e.  a  -.  c R b )
161, 15nfxfr 1692 1  |-  F/ x  R  Fr  A
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370   A.wal 1435   F/wnf 1663   F/_wnfc 2570    =/= wne 2618   A.wral 2775   E.wrex 2776    C_ wss 3436   (/)c0 3761   class class class wbr 4420    Fr wfr 4805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-br 4421  df-fr 4808
This theorem is referenced by:  nfwe  4825
  Copyright terms: Public domain W3C validator