Theorem cnvti 7085

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 4847	. . . . 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 4847	. . . . . 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 2167   class class class wbr 4033   `'ccnv 4662

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-cnv 4671

This theorem is referenced by:  eqinfti  7086  infvalti  7088  infclti  7089  inflbti  7090  infglbti  7091  infmoti  7094  infsnti  7096  infisoti  7098  infrenegsupex  9668  infxrnegsupex  11428