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Theorem cnvti 6872
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 264 . . 3  |-  ( ( -.  u R v  /\  -.  v R u )  <->  ( -.  v R u  /\  -.  u R v ) )
31, 2syl6bb 195 . 2  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( u  =  v  <-> 
( -.  v R u  /\  -.  u R v ) ) )
4 brcnvg 4688 . . . . 5  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( u `' R
v  <->  v R u ) )
54notbid 639 . . . 4  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( -.  u `' R v  <->  -.  v R u ) )
6 brcnvg 4688 . . . . . 6  |-  ( ( v  e.  A  /\  u  e.  A )  ->  ( v `' R u 
<->  u R v ) )
76ancoms 266 . . . . 5  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( v `' R u 
<->  u R v ) )
87notbid 639 . . . 4  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( -.  v `' R u  <->  -.  u R v ) )
95, 8anbi12d 462 . . 3  |-  ( ( u  e.  A  /\  v  e.  A )  ->  ( ( -.  u `' R v  /\  -.  v `' R u )  <->  ( -.  v R u  /\  -.  u R v ) ) )
109adantl 273 . 2  |-  ( (
ph  /\  ( u  e.  A  /\  v  e.  A ) )  -> 
( ( -.  u `' R v  /\  -.  v `' R u )  <->  ( -.  v R u  /\  -.  u R v ) ) )
113, 10bitr4d 190 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 103    <-> wb 104    e. wcel 1463   class class class wbr 3897   `'ccnv 4506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515
This theorem is referenced by:  eqinfti  6873  infvalti  6875  infclti  6876  inflbti  6877  infglbti  6878  infmoti  6881  infsnti  6883  infisoti  6885  infrenegsupex  9338  infxrnegsupex  10972
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