Theorem dfandc 896

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 646. (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 646	. . . 4 |- ( ph -> ( ps -> -. ( ph -> -. ps ) ) )
2	1	imp 124	. . 3 |- ( ( ph /\ ps ) -> -. ( ph -> -. ps ) )
3		simplimdc 872	. . . . . . 7 |- (DECID ph -> ( -. ( ph -> -. ps ) -> ph ) )
4	3	adantr 276	. . . . . 6 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph -> -. ps ) -> ph ) )
5	4	imp 124	. . . . 5 |- ( ( (DECID ph /\ DECID ps ) /\ -. ( ph -> -. ps ) ) -> ph )
6		simprimdc 871	. . . . . . 7 |- (DECID ps -> ( -. ( ph -> -. ps ) -> ps ) )
7	6	adantl 277	. . . . . 6 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph -> -. ps ) -> ps ) )
8	7	imp 124	. . . . 5 |- ( ( (DECID ph /\ DECID ps ) /\ -. ( ph -> -. ps ) ) -> ps )
9	5, 8	jca 306	. . . 4 |- ( ( (DECID ph /\ DECID ps ) /\ -. ( ph -> -. ps ) ) -> ( ph /\ ps ) )
10	9	ex 115	. . 3 |- ( (DECID ph /\ DECID ps ) -> ( -. ( ph -> -. ps ) -> ( ph /\ ps ) ) )
11	2, 10	impbid2 143	. 2 |- ( (DECID ph /\ DECID ps ) -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) )
12	11	ex 115	1 |- (DECID ph -> (DECID ps -> ( ( ph /\ ps ) <-> -. ( ph -> -. ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  DECID wdc 846

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This theorem depends on definitions:  df-bi 117  df-stab 843  df-dc 847

This theorem is referenced by:  pm4.63dc  898  pm4.54dc  914