Theorem ifpan23 37804

Description: Conjunction of conditional logical operators. (Contributed by RP, 20-Apr-2020.)

Assertion

Ref	Expression
ifpan23	|- ( (if- ( ph , ps , ch ) /\ if- ( ph , th , ta ) ) <-> if- ( ph , ( ps /\ th ) , ( ch /\ ta ) ) )

Proof of Theorem ifpan23

Step	Hyp	Ref	Expression
1		ifpan123g 37803	. 2 |- ( (if- ( ph , ps , ch ) /\ if- ( ph , th , ta ) ) <-> ( ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) /\ ( ( -. ph \/ th ) /\ ( ph \/ ta ) ) ) )
2		an4 865	. 2 |- ( ( ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) /\ ( ( -. ph \/ th ) /\ ( ph \/ ta ) ) ) <-> ( ( ( -. ph \/ ps ) /\ ( -. ph \/ th ) ) /\ ( ( ph \/ ch ) /\ ( ph \/ ta ) ) ) )
3		dfifp4 1016	. . 3 |- (if- ( ph , ( ps /\ th ) , ( ch /\ ta ) ) <-> ( ( -. ph \/ ( ps /\ th ) ) /\ ( ph \/ ( ch /\ ta ) ) ) )
4		ordi 908	. . . 4 |- ( ( -. ph \/ ( ps /\ th ) ) <-> ( ( -. ph \/ ps ) /\ ( -. ph \/ th ) ) )
5		ordi 908	. . . 4 |- ( ( ph \/ ( ch /\ ta ) ) <-> ( ( ph \/ ch ) /\ ( ph \/ ta ) ) )
6	4, 5	anbi12i 733	. . 3 |- ( ( ( -. ph \/ ( ps /\ th ) ) /\ ( ph \/ ( ch /\ ta ) ) ) <-> ( ( ( -. ph \/ ps ) /\ ( -. ph \/ th ) ) /\ ( ( ph \/ ch ) /\ ( ph \/ ta ) ) ) )
7	3, 6	bitr2i 265	. 2 |- ( ( ( ( -. ph \/ ps ) /\ ( -. ph \/ th ) ) /\ ( ( ph \/ ch ) /\ ( ph \/ ta ) ) ) <-> if- ( ph , ( ps /\ th ) , ( ch /\ ta ) ) )
8	1, 2, 7	3bitri 286	1 |- ( (if- ( ph , ps , ch ) /\ if- ( ph , th , ta ) ) <-> if- ( ph , ( ps /\ th ) , ( ch /\ ta ) ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ 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:  ifpdfbi  37818  ifpdfxor  37832