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Theorem cnvti 6619
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 4573 . . . . 5  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( u `' R
v  <->  v R u ) )
54notbid 625 . . . 4  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( -.  u `' R v  <->  -.  v R u ) )
6 brcnvg 4573 . . . . . 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 625 . . . 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 1434   class class class wbr 3811   `'ccnv 4398
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-cnv 4407
This theorem is referenced by:  eqinfti  6620  infvalti  6622  infclti  6623  inflbti  6624  infglbti  6625  infmoti  6628  infsnti  6630  infisoti  6632  infrenegsupex  8975
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