Theorem ifpbibib 37855

Description: Factor conditional logic operator over biimplication in terms 2 and 3. (Contributed by RP, 21-Apr-2020.)

Assertion

Ref	Expression
ifpbibib	|- (if- ( ph , ( ps <-> ch ) , ( th <-> ta ) ) <-> (if- ( ph , ps , th ) <-> if- ( ph , ch , ta ) ) )

Proof of Theorem ifpbibib

Step	Hyp	Ref	Expression
1		dfifp2 1014	. 2 |- (if- ( ph , ( ps <-> ch ) , ( th <-> ta ) ) <-> ( ( ph -> ( ps <-> ch ) ) /\ ( -. ph -> ( th <-> ta ) ) ) )
2		dfbi2 660	. . . . . 6 |- ( ( ps <-> ch ) <-> ( ( ps -> ch ) /\ ( ch -> ps ) ) )
3	2	imbi2i 326	. . . . 5 |- ( ( ph -> ( ps <-> ch ) ) <-> ( ph -> ( ( ps -> ch ) /\ ( ch -> ps ) ) ) )
4		jcab 907	. . . . 5 |- ( ( ph -> ( ( ps -> ch ) /\ ( ch -> ps ) ) ) <-> ( ( ph -> ( ps -> ch ) ) /\ ( ph -> ( ch -> ps ) ) ) )
5	3, 4	bitri 264	. . . 4 |- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph -> ( ps -> ch ) ) /\ ( ph -> ( ch -> ps ) ) ) )
6		dfbi2 660	. . . . . 6 |- ( ( th <-> ta ) <-> ( ( th -> ta ) /\ ( ta -> th ) ) )
7	6	imbi2i 326	. . . . 5 |- ( ( -. ph -> ( th <-> ta ) ) <-> ( -. ph -> ( ( th -> ta ) /\ ( ta -> th ) ) ) )
8		jcab 907	. . . . 5 |- ( ( -. ph -> ( ( th -> ta ) /\ ( ta -> th ) ) ) <-> ( ( -. ph -> ( th -> ta ) ) /\ ( -. ph -> ( ta -> th ) ) ) )
9	7, 8	bitri 264	. . . 4 |- ( ( -. ph -> ( th <-> ta ) ) <-> ( ( -. ph -> ( th -> ta ) ) /\ ( -. ph -> ( ta -> th ) ) ) )
10	5, 9	anbi12i 733	. . 3 |- ( ( ( ph -> ( ps <-> ch ) ) /\ ( -. ph -> ( th <-> ta ) ) ) <-> ( ( ( ph -> ( ps -> ch ) ) /\ ( ph -> ( ch -> ps ) ) ) /\ ( ( -. ph -> ( th -> ta ) ) /\ ( -. ph -> ( ta -> th ) ) ) ) )
11		an4 865	. . 3 |- ( ( ( ( ph -> ( ps -> ch ) ) /\ ( ph -> ( ch -> ps ) ) ) /\ ( ( -. ph -> ( th -> ta ) ) /\ ( -. ph -> ( ta -> th ) ) ) ) <-> ( ( ( ph -> ( ps -> ch ) ) /\ ( -. ph -> ( th -> ta ) ) ) /\ ( ( ph -> ( ch -> ps ) ) /\ ( -. ph -> ( ta -> th ) ) ) ) )
12	10, 11	bitri 264	. 2 |- ( ( ( ph -> ( ps <-> ch ) ) /\ ( -. ph -> ( th <-> ta ) ) ) <-> ( ( ( ph -> ( ps -> ch ) ) /\ ( -. ph -> ( th -> ta ) ) ) /\ ( ( ph -> ( ch -> ps ) ) /\ ( -. ph -> ( ta -> th ) ) ) ) )
13		dfifp2 1014	. . . . 5 |- (if- ( ph , ( ps -> ch ) , ( th -> ta ) ) <-> ( ( ph -> ( ps -> ch ) ) /\ ( -. ph -> ( th -> ta ) ) ) )
14		ifpimimb 37849	. . . . 5 |- (if- ( ph , ( ps -> ch ) , ( th -> ta ) ) <-> (if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) )
15	13, 14	bitr3i 266	. . . 4 |- ( ( ( ph -> ( ps -> ch ) ) /\ ( -. ph -> ( th -> ta ) ) ) <-> (if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) )
16		dfifp2 1014	. . . . 5 |- (if- ( ph , ( ch -> ps ) , ( ta -> th ) ) <-> ( ( ph -> ( ch -> ps ) ) /\ ( -. ph -> ( ta -> th ) ) ) )
17		ifpimimb 37849	. . . . 5 |- (if- ( ph , ( ch -> ps ) , ( ta -> th ) ) <-> (if- ( ph , ch , ta ) -> if- ( ph , ps , th ) ) )
18	16, 17	bitr3i 266	. . . 4 |- ( ( ( ph -> ( ch -> ps ) ) /\ ( -. ph -> ( ta -> th ) ) ) <-> (if- ( ph , ch , ta ) -> if- ( ph , ps , th ) ) )
19	15, 18	anbi12i 733	. . 3 |- ( ( ( ( ph -> ( ps -> ch ) ) /\ ( -. ph -> ( th -> ta ) ) ) /\ ( ( ph -> ( ch -> ps ) ) /\ ( -. ph -> ( ta -> th ) ) ) ) <-> ( (if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) /\ (if- ( ph , ch , ta ) -> if- ( ph , ps , th ) ) ) )
20		dfbi2 660	. . 3 |- ( (if- ( ph , ps , th ) <-> if- ( ph , ch , ta ) ) <-> ( (if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) /\ (if- ( ph , ch , ta ) -> if- ( ph , ps , th ) ) ) )
21	19, 20	bitr4i 267	. 2 |- ( ( ( ( ph -> ( ps -> ch ) ) /\ ( -. ph -> ( th -> ta ) ) ) /\ ( ( ph -> ( ch -> ps ) ) /\ ( -. ph -> ( ta -> th ) ) ) ) <-> (if- ( ph , ps , th ) <-> if- ( ph , ch , ta ) ) )
22	1, 12, 21	3bitri 286	1 |- (if- ( ph , ( ps <-> ch ) , ( th <-> ta ) ) <-> (if- ( ph , ps , th ) <-> if- ( ph , ch , ta ) ) )

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

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

This theorem is referenced by:  ifpxorxorb  37856