Theorem anxordi 1390

Description: Conjunction distributes over exclusive-or. (Contributed by Mario Carneiro and Jim Kingdon, 7-Oct-2018.)

Assertion

Ref	Expression
anxordi	|- ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) )

Proof of Theorem anxordi

Step	Hyp	Ref	Expression
1		simpl 108	. 2 |- ( ( ph /\ ( ps \/_ ch ) ) -> ph )
2		df-xor 1366	. . . 4 |- ( ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) <-> ( ( ( ph /\ ps ) \/ ( ph /\ ch ) ) /\ -. ( ( ph /\ ps ) /\ ( ph /\ ch ) ) ) )
3	2	simplbi 272	. . 3 |- ( ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
4		simpl 108	. . . 4 |- ( ( ph /\ ps ) -> ph )
5		simpl 108	. . . 4 |- ( ( ph /\ ch ) -> ph )
6	4, 5	jaoi 706	. . 3 |- ( ( ( ph /\ ps ) \/ ( ph /\ ch ) ) -> ph )
7	3, 6	syl 14	. 2 |- ( ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) -> ph )
8		ibar 299	. . 3 |- ( ph -> ( ( ps \/_ ch ) <-> ( ph /\ ( ps \/_ ch ) ) ) )
9		ibar 299	. . . 4 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
10		ibar 299	. . . 4 |- ( ph -> ( ch <-> ( ph /\ ch ) ) )
11	9, 10	xorbi12d 1372	. . 3 |- ( ph -> ( ( ps \/_ ch ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) )
12	8, 11	bitr3d 189	. 2 |- ( ph -> ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) )
13	1, 7, 12	pm5.21nii 694	1 |- ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) )

Syntax hints:   -. wn 3   /\ wa 103   <-> wb 104   \/ wo 698   \/_ wxo 1365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-xor 1366

This theorem is referenced by:  rpnegap  9622