Theorem cnvti 7078

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 266	. . 3 |- ( ( -. u R v /\ -. v R u ) <-> ( -. v R u /\ -. u R v ) )
3	1, 2	bitrdi 196	. 2 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. v R u /\ -. u R v ) ) )
4		brcnvg 4843	. . . . 5 |- ( ( u e. A /\ v e. A ) -> ( u `' R v <-> v R u ) )
5	4	notbid 668	. . . 4 |- ( ( u e. A /\ v e. A ) -> ( -. u `' R v <-> -. v R u ) )
6		brcnvg 4843	. . . . . 6 |- ( ( v e. A /\ u e. A ) -> ( v `' R u <-> u R v ) )
7	6	ancoms 268	. . . . 5 |- ( ( u e. A /\ v e. A ) -> ( v `' R u <-> u R v ) )
8	7	notbid 668	. . . 4 |- ( ( u e. A /\ v e. A ) -> ( -. v `' R u <-> -. u R v ) )
9	5, 8	anbi12d 473	. . 3 |- ( ( u e. A /\ v e. A ) -> ( ( -. u `' R v /\ -. v `' R u ) <-> ( -. v R u /\ -. u R v ) ) )
10	9	adantl 277	. 2 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( ( -. u `' R v /\ -. v `' R u ) <-> ( -. v R u /\ -. u R v ) ) )
11	3, 10	bitr4d 191	1 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u `' R v /\ -. v `' R u ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2164   class class class wbr 4029   `'ccnv 4658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-cnv 4667

This theorem is referenced by:  eqinfti  7079  infvalti  7081  infclti  7082  inflbti  7083  infglbti  7084  infmoti  7087  infsnti  7089  infisoti  7091  infrenegsupex  9659  infxrnegsupex  11406