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

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 262	. . 3 |- ( ( -. u R v /\ -. v R u ) <-> ( -. v R u /\ -. u R v ) )
3	1, 2	syl6bb 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 ) )
5	4	notbid 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 ) )
7	6	ancoms 264	. . . . 5 |- ( ( u e. A /\ v e. A ) -> ( v `' R u <-> u R v ) )
8	7	notbid 625	. . . 4 |- ( ( u e. A /\ v e. A ) -> ( -. v `' R u <-> -. u R v ) )
9	5, 8	anbi12d 457	. . 3 |- ( ( u e. A /\ v e. A ) -> ( ( -. u `' R v /\ -. v `' R u ) <-> ( -. v R u /\ -. u R v ) ) )
10	9	adantl 271	. 2 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( ( -. u `' R v /\ -. v `' R u ) <-> ( -. v R u /\ -. u R v ) ) )
11	3, 10	bitr4d 189	1 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u `' R v /\ -. v `' R u ) ) )

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