Theorem nbbndc 1325

Description: Move negation outside of biconditional, for decidable propositions. Compare Theorem *5.18 of [WhiteheadRussell] p. 124. (Contributed by Jim Kingdon, 18-Apr-2018.)

Assertion

Ref	Expression
nbbndc	|- (DECID ph -> (DECID ps -> ( ( -. ph <-> ps ) <-> -. ( ph <-> ps ) ) ) )

Proof of Theorem nbbndc

Step	Hyp	Ref	Expression
1		xor3dc 1318	. . . . 5 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) ) )
2	1	imp 122	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) )
3		con2bidc 802	. . . . 5 |- (DECID ph -> (DECID ps -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) ) )
4	3	imp 122	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) )
5	2, 4	bitrd 186	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ps <-> -. ph ) ) )
6		bicom 138	. . 3 |- ( ( ps <-> -. ph ) <-> ( -. ph <-> ps ) )
7	5, 6	syl6rbb 195	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph <-> ps ) <-> -. ( ph <-> ps ) ) )
8	7	ex 113	1 |- (DECID ph -> (DECID ps -> ( ( -. ph <-> ps ) <-> -. ( ph <-> ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103  DECID wdc 775

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

This theorem is referenced by:  biassdc  1326