Theorem ifpdfbi 37818

Description: Define biimplication as conditional logic operator. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpdfbi	|- ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) )

Proof of Theorem ifpdfbi

Step	Hyp	Ref	Expression
1		dfbi2 660	. 2 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2		ifpim1 37813	. . . . 5 |- ( ( ph -> ps ) <-> if- ( -. ph , T. , ps ) )
3		ifpn 1022	. . . . 5 |- (if- ( ph , ps , T. ) <-> if- ( -. ph , T. , ps ) )
4	2, 3	bitr4i 267	. . . 4 |- ( ( ph -> ps ) <-> if- ( ph , ps , T. ) )
5		ifpim2 37816	. . . 4 |- ( ( ps -> ph ) <-> if- ( ph , T. , -. ps ) )
6	4, 5	anbi12i 733	. . 3 |- ( ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> (if- ( ph , ps , T. ) /\ if- ( ph , T. , -. ps ) ) )
7		ifpan23 37804	. . . 4 |- ( (if- ( ph , ps , T. ) /\ if- ( ph , T. , -. ps ) ) <-> if- ( ph , ( ps /\ T. ) , ( T. /\ -. ps ) ) )
8		ancom 466	. . . . . 6 |- ( ( ps /\ T. ) <-> ( T. /\ ps ) )
9		truan 1501	. . . . . 6 |- ( ( T. /\ ps ) <-> ps )
10	8, 9	bitri 264	. . . . 5 |- ( ( ps /\ T. ) <-> ps )
11		truan 1501	. . . . 5 |- ( ( T. /\ -. ps ) <-> -. ps )
12		ifpbi23 37817	. . . . 5 |- ( ( ( ( ps /\ T. ) <-> ps ) /\ ( ( T. /\ -. ps ) <-> -. ps ) ) -> (if- ( ph , ( ps /\ T. ) , ( T. /\ -. ps ) ) <-> if- ( ph , ps , -. ps ) ) )
13	10, 11, 12	mp2an 708	. . . 4 |- (if- ( ph , ( ps /\ T. ) , ( T. /\ -. ps ) ) <-> if- ( ph , ps , -. ps ) )
14	7, 13	bitri 264	. . 3 |- ( (if- ( ph , ps , T. ) /\ if- ( ph , T. , -. ps ) ) <-> if- ( ph , ps , -. ps ) )
15	6, 14	bitri 264	. 2 |- ( ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> if- ( ph , ps , -. ps ) )
16	1, 15	bitri 264	1 |- ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384  if-wif 1012   T. wtru 1484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-tru 1486

This theorem is referenced by:  ifpbiidcor  37819  ifpbicor  37820