Theorem cnvti 6955

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

Step	Hyp	Ref	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 ) )
3	1, 2	bitrdi 195	. 2 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. v R u /\ -. u R v ) ) )
4		brcnvg 4764	. . . . 5 |- ( ( u e. A /\ v e. A ) -> ( u `' R v <-> v R u ) )
5	4	notbid 657	. . . 4 |- ( ( u e. A /\ v e. A ) -> ( -. u `' R v <-> -. v R u ) )
6		brcnvg 4764	. . . . . 6 |- ( ( v e. A /\ u e. A ) -> ( v `' R u <-> u R v ) )
7	6	ancoms 266	. . . . 5 |- ( ( u e. A /\ v e. A ) -> ( v `' R u <-> u R v ) )
8	7	notbid 657	. . . 4 |- ( ( u e. A /\ v e. A ) -> ( -. v `' R u <-> -. u R v ) )
9	5, 8	anbi12d 465	. . 3 |- ( ( u e. A /\ v e. A ) -> ( ( -. u `' R v /\ -. v `' R u ) <-> ( -. v R u /\ -. u R v ) ) )
10	9	adantl 275	. 2 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( ( -. u `' R v /\ -. v `' R u ) <-> ( -. v R u /\ -. u R v ) ) )
11	3, 10	bitr4d 190	1 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u `' R v /\ -. v `' R u ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2128   class class class wbr 3965   `'ccnv 4582

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4591

This theorem is referenced by:  eqinfti  6956  infvalti  6958  infclti  6959  inflbti  6960  infglbti  6961  infmoti  6964  infsnti  6966  infisoti  6968  infrenegsupex  9488  infxrnegsupex  11142