Theorem xordc1 1324

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 765	. . 3 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) <-> ( ( ph /\ -. ( ph /\ ps ) ) \/ ( ps /\ -. ( ph /\ ps ) ) ) )
2		simpl 107	. . . 4 |- ( ( ph /\ -. ( ph /\ ps ) ) -> ph )
3		imnan 656	. . . . . 6 |- ( ( ps -> -. ph ) <-> -. ( ps /\ ph ) )
4		ancom 262	. . . . . 6 |- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
5	3, 4	xchbinxr 640	. . . . 5 |- ( ( ps -> -. ph ) <-> -. ( ph /\ ps ) )
6		pm3.35 339	. . . . 5 |- ( ( ps /\ ( ps -> -. ph ) ) -> -. ph )
7	5, 6	sylan2br 282	. . . 4 |- ( ( ps /\ -. ( ph /\ ps ) ) -> -. ph )
8	2, 7	orim12i 708	. . 3 |- ( ( ( ph /\ -. ( ph /\ ps ) ) \/ ( ps /\ -. ( ph /\ ps ) ) ) -> ( ph \/ -. ph ) )
9	1, 8	sylbi 119	. 2 |- ( ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) -> ( ph \/ -. ph ) )
10		df-xor 1307	. 2 |- ( ( ph \/_ ps ) <-> ( ( ph \/ ps ) /\ -. ( ph /\ ps ) ) )
11		df-dc 776	. 2 |- (DECID ph <-> ( ph \/ -. ph ) )
12	9, 10, 11	3imtr4i 199	1 |- ( ( ph \/_ ps ) -> DECID ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661  DECID wdc 775   \/_ wxo 1306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776  df-xor 1307

This theorem is referenced by: (None)