Theorem xordc1 1388

Description: Exclusive or implies the left proposition is decidable. (Contributed by Jim Kingdon, 12-Mar-2018.)

Assertion

Ref	Expression
xordc1	|- ( ( ph \/_ ps ) -> DECID ph )

Proof of Theorem xordc1

Step	Hyp	Ref	Expression
1		andir 814	. . 3 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ( ph /\ -. ( ph /\ ps ) ) \/ ( ps /\ -. ( ph /\ ps ) ) ) )
2		simpl 108	. . . 4 |- ( ( ph /\ -. ( ph /\ ps ) ) -> ph )
3		imnan 685	. . . . . 6 |- ( ( ps -> -. ph ) <-> -. ( ps /\ ph ) )
4		ancom 264	. . . . . 6 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
5	3, 4	xchbinxr 678	. . . . 5 |- ( ( ps -> -. ph ) <-> -. ( ph /\ ps ) )
6		pm3.35 345	. . . . 5 |- ( ( ps /\ ( ps -> -. ph ) ) -> -. ph )
7	5, 6	sylan2br 286	. . . 4 |- ( ( ps /\ -. ( ph /\ ps ) ) -> -. ph )
8	2, 7	orim12i 754	. . 3 |- ( ( ( ph /\ -. ( ph /\ ps ) ) \/ ( ps /\ -. ( ph /\ ps ) ) ) -> ( ph \/ -. ph ) )
9	1, 8	sylbi 120	. 2 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) -> ( ph \/ -. ph ) )
10		df-xor 1371	. 2 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
11		df-dc 830	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
12	9, 10, 11	3imtr4i 200	1 |- ( ( ph \/_ ps ) -> DECID ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 703  DECID wdc 829   \/_ wxo 1370

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 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-dc 830  df-xor 1371

This theorem is referenced by: (None)