Theorem pm5.54dc 827

Description: A conjunction is equivalent to one of its conjuncts, given a decidable conjunct. Based on theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by Jim Kingdon, 30-Mar-2018.)

Assertion

Ref	Expression
pm5.54dc	|- (DECID ph -> ( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) ) )

Proof of Theorem pm5.54dc

Step	Hyp	Ref	Expression
1		df-dc 743	. . 3 |- (DECID ph <-> ( ph \/ -. ph ) )
2		simpr 103	. . . . 5 |- ( ( ph /\ ps ) -> ps )
3		ax-ia3 101	. . . . 5 |- ( ph -> ( ps -> ( ph /\ ps ) ) )
4	2, 3	impbid2 131	. . . 4 |- ( ph -> ( ( ph /\ ps ) <-> ps ) )
5		simpl 102	. . . . 5 |- ( ( ph /\ ps ) -> ph )
6		ax-in2 545	. . . . 5 |- ( -. ph -> ( ph -> ( ph /\ ps ) ) )
7	5, 6	impbid2 131	. . . 4 |- ( -. ph -> ( ( ph /\ ps ) <-> ph ) )
8	4, 7	orim12i 676	. . 3 |- ( ( ph \/ -. ph ) -> ( ( ( ph /\ ps ) <-> ps ) \/ ( ( ph /\ ps ) <-> ph ) ) )
9	1, 8	sylbi 114	. 2 |- (DECID ph -> ( ( ( ph /\ ps ) <-> ps ) \/ ( ( ph /\ ps ) <-> ph ) ) )
10	9	orcomd 648	1 |- (DECID ph -> ( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629  DECID wdc 742

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-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)