Theorem anxordi 1291

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 102	. 2 |- ( ( ph /\ ( ps \/_ ch ) ) -> ph )
2		df-xor 1267	. . . 4 |- ( ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) <-> ( ( ( ph /\ ps ) \/ ( ph /\ ch ) ) /\ -. ( ( ph /\ ps ) /\ ( ph /\ ch ) ) ) )
3	2	simplbi 259	. . 3 |- ( ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) -> ( ( ph /\ ps ) \/ ( ph /\ ch ) ) )
4		simpl 102	. . . 4 |- ( ( ph /\ ps ) -> ph )
5		simpl 102	. . . 4 |- ( ( ph /\ ch ) -> ph )
6	4, 5	jaoi 636	. . 3 |- ( ( ( ph /\ ps ) \/ ( ph /\ ch ) ) -> ph )
7	3, 6	syl 14	. 2 |- ( ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) -> ph )
8		ibar 285	. . 3 |- ( ph -> ( ( ps \/_ ch ) <-> ( ph /\ ( ps \/_ ch ) ) ) )
9		ibar 285	. . . 4 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
10		ibar 285	. . . 4 |- ( ph -> ( ch <-> ( ph /\ ch ) ) )
11	9, 10	xorbi12d 1273	. . 3 |- ( ph -> ( ( ps \/_ ch ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) )
12	8, 11	bitr3d 179	. 2 |- ( ph -> ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) ) )
13	1, 7, 12	pm5.21nii 620	1 |- ( ( ph /\ ( ps \/_ ch ) ) <-> ( ( ph /\ ps ) \/_ ( ph /\ ch ) ) )

Syntax hints:   -. wn 3   /\ wa 97   <-> wb 98   \/ wo 629   \/_ wxo 1266

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-xor 1267

This theorem is referenced by:  rpnegap  8615