Theorem pm5.63dc 911

Description: Theorem *5.63 of [WhiteheadRussell] p. 125, for a decidable proposition. (Contributed by Jim Kingdon, 12-May-2018.)

Assertion

Ref	Expression
pm5.63dc	|- (DECID ph -> ( ( ph \/ ps ) <-> ( ph \/ ( -. ph /\ ps ) ) ) )

Proof of Theorem pm5.63dc

Step	Hyp	Ref	Expression
1		df-dc 803	. . 3 |- (DECID ph <-> ( ph \/ -. ph ) )
2		ordi 788	. . . 4 |- ( ( ph \/ ( -. ph /\ ps ) ) <-> ( ( ph \/ -. ph ) /\ ( ph \/ ps ) ) )
3	2	simplbi2 380	. . 3 |- ( ( ph \/ -. ph ) -> ( ( ph \/ ps ) -> ( ph \/ ( -. ph /\ ps ) ) ) )
4	1, 3	sylbi 120	. 2 |- (DECID ph -> ( ( ph \/ ps ) -> ( ph \/ ( -. ph /\ ps ) ) ) )
5	2	simprbi 271	. 2 |- ( ( ph \/ ( -. ph /\ ps ) ) -> ( ph \/ ps ) )
6	4, 5	impbid1 141	1 |- (DECID ph -> ( ( ph \/ ps ) <-> ( ph \/ ( -. ph /\ ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 680  DECID wdc 802

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-io 681

This theorem depends on definitions:  df-bi 116  df-dc 803

This theorem is referenced by: (None)