Theorem dfandc 852

Description: Definition of 'and' in terms of negation and implication, for decidable propositions. The forward direction holds for all propositions, and can (basically) be found at pm3.2im 609. (Contributed by Jim Kingdon, 30-Apr-2018.)

Assertion

Ref	Expression
dfandc	|- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) ) )

Proof of Theorem dfandc

Step	Hyp	Ref	Expression
1		pm3.2im 609	. . . 4 |- ( ph -> ( ps -> -. ( ph -> -. ps ) ) )
2	1	imp 123	. . 3 |- ( ( ph /\ ps ) -> -. ( ph -> -. ps ) )
3		simplimdc 828	. . . . . . 7 |- (DECID ph -> ( -. ( ph -> -. ps ) -> ph ) )
4	3	adantr 272	. . . . . 6 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph -> -. ps ) -> ph ) )
5	4	imp 123	. . . . 5 |- ( ( (DECID ph /\ DECID ps ) /\ -. ( ph -> -. ps ) ) -> ph )
6		simprimdc 827	. . . . . . 7 |- (DECID ps -> ( -. ( ph -> -. ps ) -> ps ) )
7	6	adantl 273	. . . . . 6 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph -> -. ps ) -> ps ) )
8	7	imp 123	. . . . 5 |- ( ( (DECID ph /\ DECID ps ) /\ -. ( ph -> -. ps ) ) -> ps )
9	5, 8	jca 302	. . . 4 |- ( ( (DECID ph /\ DECID ps ) /\ -. ( ph -> -. ps ) ) -> ( ph /\ ps ) )
10	9	ex 114	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph -> -. ps ) -> ( ph /\ ps ) ) )
11	2, 10	impbid2 142	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) )
12	11	ex 114	1 |- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104  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-in1 586  ax-in2 587  ax-io 681

This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803

This theorem is referenced by:  pm4.63dc  854  pm4.54dc  870