Theorem nbbndc 1389

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 1382	. . . . 5 |- (DECID ph -> (DECID ps -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) ) )
2	1	imp 123	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ph <-> -. ps ) ) )
3		con2bidc 870	. . . . 5 |- (DECID ph -> (DECID ps -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) ) )
4	3	imp 123	. . . 4 |- ( (DECID ph /\ DECID ps ) -> ( ( ph <-> -. ps ) <-> ( ps <-> -. ph ) ) )
5	2, 4	bitrd 187	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph <-> ps ) <-> ( ps <-> -. ph ) ) )
6		bicom 139	. . 3 |- ( ( ps <-> -. ph ) <-> ( -. ph <-> ps ) )
7	5, 6	bitr2di 196	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( -. ph <-> ps ) <-> -. ( ph <-> ps ) ) )
8	7	ex 114	1 |- (DECID ph -> (DECID ps -> ( ( -. ph <-> ps ) <-> -. ( ph <-> ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  DECID wdc 829

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-stab 826  df-dc 830

This theorem is referenced by:  biassdc  1390