ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfwe Unicode version

Theorem nfwe 4120
Description: Bound-variable hypothesis builder for well-orderings. (Contributed by Stefan O'Rear, 20-Jan-2015.) (Revised by Mario Carneiro, 14-Oct-2016.)
Hypotheses
Ref Expression
nfwe.r  |-  F/_ x R
nfwe.a  |-  F/_ x A
Assertion
Ref Expression
nfwe  |-  F/ x  R  We  A

Proof of Theorem nfwe
Dummy variables  a  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wetr 4099 . 2  |-  ( R  We  A  <->  ( R  Fr  A  /\  A. a  e.  A  A. b  e.  A  A. c  e.  A  ( (
a R b  /\  b R c )  -> 
a R c ) ) )
2 nfwe.r . . . 4  |-  F/_ x R
3 nfwe.a . . . 4  |-  F/_ x A
42, 3nffr 4114 . . 3  |-  F/ x  R  Fr  A
5 nfcv 2194 . . . . . . . . 9  |-  F/_ x
a
6 nfcv 2194 . . . . . . . . 9  |-  F/_ x
b
75, 2, 6nfbr 3836 . . . . . . . 8  |-  F/ x  a R b
8 nfcv 2194 . . . . . . . . 9  |-  F/_ x
c
96, 2, 8nfbr 3836 . . . . . . . 8  |-  F/ x  b R c
107, 9nfan 1473 . . . . . . 7  |-  F/ x
( a R b  /\  b R c )
115, 2, 8nfbr 3836 . . . . . . 7  |-  F/ x  a R c
1210, 11nfim 1480 . . . . . 6  |-  F/ x
( ( a R b  /\  b R c )  ->  a R c )
133, 12nfralxy 2377 . . . . 5  |-  F/ x A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
143, 13nfralxy 2377 . . . 4  |-  F/ x A. b  e.  A  A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
153, 14nfralxy 2377 . . 3  |-  F/ x A. a  e.  A  A. b  e.  A  A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c )
164, 15nfan 1473 . 2  |-  F/ x
( R  Fr  A  /\  A. a  e.  A  A. b  e.  A  A. c  e.  A  ( ( a R b  /\  b R c )  ->  a R c ) )
171, 16nfxfr 1379 1  |-  F/ x  R  We  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   F/wnf 1365   F/_wnfc 2181   A.wral 2323   class class class wbr 3792    Fr wfr 4093    We wwe 4095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-frfor 4096  df-frind 4097  df-wetr 4099
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator