Theorem ifpxorxorb 37856

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

Assertion

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

Proof of Theorem ifpxorxorb

Step	Hyp	Ref	Expression
1		df-xor 1465	. . 3 |- ( ( ps \/_ ch ) <-> -. ( ps <-> ch ) )
2		df-xor 1465	. . 3 |- ( ( th \/_ ta ) <-> -. ( th <-> ta ) )
3		ifpbi23 37817	. . 3 |- ( ( ( ( ps \/_ ch ) <-> -. ( ps <-> ch ) ) /\ ( ( th \/_ ta ) <-> -. ( th <-> ta ) ) ) -> (if- ( ph , ( ps \/_ ch ) , ( th \/_ ta ) ) <-> if- ( ph , -. ( ps <-> ch ) , -. ( th <-> ta ) ) ) )
4	1, 2, 3	mp2an 708	. 2 |- (if- ( ph , ( ps \/_ ch ) , ( th \/_ ta ) ) <-> if- ( ph , -. ( ps <-> ch ) , -. ( th <-> ta ) ) )
5		ifpbibib 37855	. . . 4 |- (if- ( ph , ( ps <-> ch ) , ( th <-> ta ) ) <-> (if- ( ph , ps , th ) <-> if- ( ph , ch , ta ) ) )
6	5	notbii 310	. . 3 |- ( -. if- ( ph , ( ps <-> ch ) , ( th <-> ta ) ) <-> -. (if- ( ph , ps , th ) <-> if- ( ph , ch , ta ) ) )
7		ifpnotnotb 37824	. . 3 |- (if- ( ph , -. ( ps <-> ch ) , -. ( th <-> ta ) ) <-> -. if- ( ph , ( ps <-> ch ) , ( th <-> ta ) ) )
8		df-xor 1465	. . 3 |- ( (if- ( ph , ps , th ) \/_ if- ( ph , ch , ta ) ) <-> -. (if- ( ph , ps , th ) <-> if- ( ph , ch , ta ) ) )
9	6, 7, 8	3bitr4i 292	. 2 |- (if- ( ph , -. ( ps <-> ch ) , -. ( th <-> ta ) ) <-> (if- ( ph , ps , th ) \/_ if- ( ph , ch , ta ) ) )
10	4, 9	bitri 264	1 |- (if- ( ph , ( ps \/_ ch ) , ( th \/_ ta ) ) <-> (if- ( ph , ps , th ) \/_ if- ( ph , ch , ta ) ) )

Syntax hints:   -. wn 3   <-> wb 196  if-wif 1012   \/_ wxo 1464

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-xor 1465

This theorem is referenced by: (None)