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Theorem cnvti 6693
Description: If a relation satisfies a condition corresponding to tightness of an apartness generated by an order, so does its converse. (Contributed by Jim Kingdon, 17-Dec-2021.)
Hypothesis
Ref Expression
eqinfti.ti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
Assertion
Ref Expression
cnvti  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u `' R v  /\  -.  v `' R u ) ) )
Distinct variable groups:    u, A, v    ph, u, v    u, R, v

Proof of Theorem cnvti
StepHypRef Expression
1 eqinfti.ti . . 3  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u R v  /\  -.  v R u ) ) )
2 ancom 262 . . 3  |-  ( ( -.  u R v  /\  -.  v R u )  <->  ( -.  v R u  /\  -.  u R v ) )
31, 2syl6bb 194 . 2  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  v R u  /\  -.  u R v ) ) )
4 brcnvg 4605 . . . . 5  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( u `' R
v  <->  v R u ) )
54notbid 627 . . . 4  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( -.  u `' R v  <->  -.  v R u ) )
6 brcnvg 4605 . . . . . 6  |-  ( ( v  e.  A  /\  u  e.  A )  ->  ( v `' R u 
<->  u R v ) )
76ancoms 264 . . . . 5  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( v `' R u 
<->  u R v ) )
87notbid 627 . . . 4  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( -.  v `' R u  <->  -.  u R v ) )
95, 8anbi12d 457 . . 3  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( ( -.  u `' R v  /\  -.  v `' R u )  <->  ( -.  v R u  /\  -.  u R v ) ) )
109adantl 271 . 2  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( ( -.  u `' R v  /\  -.  v `' R u )  <->  ( -.  v R u  /\  -.  u R v ) ) )
113, 10bitr4d 189 1  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  u `' R v  /\  -.  v `' R u ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1438   class class class wbr 3837   `'ccnv 4427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-cnv 4436
This theorem is referenced by:  eqinfti  6694  infvalti  6696  infclti  6697  inflbti  6698  infglbti  6699  infmoti  6702  infsnti  6704  infisoti  6706  infrenegsupex  9051
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