Theorem xoranor 1338

Description: One way of defining exclusive or. Equivalent to df-xor 1337. (Contributed by Jim Kingdon and Mario Carneiro, 1-Mar-2018.)

Assertion

Ref	Expression
xoranor	|- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ ( -. ph \/ -. ps ) ) )

Proof of Theorem xoranor

Step	Hyp	Ref	Expression
1		df-xor 1337	. . 3 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
2		ax-ia3 107	. . . . . . 7 |- ( ph -> ( ps -> ( ph /\ ps ) ) )
3	2	con3d 603	. . . . . 6 |- ( ph -> ( -. ( ph /\ ps ) -> -. ps ) )
4		olc 683	. . . . . 6 |- ( -. ps -> ( -. ph \/ -. ps ) )
5	3, 4	syl6 33	. . . . 5 |- ( ph -> ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) )
6		pm3.21 262	. . . . . . 7 |- ( ps -> ( ph -> ( ph /\ ps ) ) )
7	6	con3d 603	. . . . . 6 |- ( ps -> ( -. ( ph /\ ps ) -> -. ph ) )
8		orc 684	. . . . . 6 |- ( -. ph -> ( -. ph \/ -. ps ) )
9	7, 8	syl6 33	. . . . 5 |- ( ps -> ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) )
10	5, 9	jaoi 688	. . . 4 |- ( ( ph \/ ps ) -> ( -. ( ph /\ ps ) -> ( -. ph \/ -. ps ) ) )
11	10	imdistani 439	. . 3 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) -> ( ( ph \/ ps ) /\ ( -. ph \/ -. ps ) ) )
12	1, 11	sylbi 120	. 2 |- ( ( ph \/_ ps ) -> ( ( ph \/ ps ) /\ ( -. ph \/ -. ps ) ) )
13		pm3.14 725	. . . 4 |- ( ( -. ph \/ -. ps ) -> -. ( ph /\ ps ) )
14	13	anim2i 337	. . 3 |- ( ( ( ph \/ ps ) /\ ( -. ph \/ -. ps ) ) -> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
15	14, 1	sylibr 133	. 2 |- ( ( ( ph \/ ps ) /\ ( -. ph \/ -. ps ) ) -> ( ph \/_ ps ) )
16	12, 15	impbii 125	1 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ ( -. ph \/ -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 680   \/_ wxo 1336

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681

This theorem depends on definitions:  df-bi 116  df-xor 1337

This theorem is referenced by:  excxor  1339  xoror  1340