Theorem anordc 862

Description: Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120, but where the propositions are decidable. The forward direction, pm3.1 670, holds for all propositions, but the equivalence only holds given decidability. (Contributed by Jim Kingdon, 21-Apr-2018.)

Assertion

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

Proof of Theorem anordc

Step	Hyp	Ref	Expression
1		dcan 841	. 2 |- (DECID ph -> (DECID ps -> DECID (ph /\ ps)))
2		ianordc 798	. . . . 5 |- (DECID ph -> (-. (ph /\ ps) <-> (-. ph \/ -. ps)))
3	2	bicomd 129	. . . 4 |- (DECID ph -> ((-. ph \/ -. ps) <-> -. (ph /\ ps)))
4	3	a1d 22	. . 3 |- (DECID ph -> (DECID (ph /\ ps) -> ((-. ph \/ -. ps) <-> -. (ph /\ ps))))
5	4	con2biddc 773	. 2 |- (DECID ph -> (DECID (ph /\ ps) -> ((ph /\ ps) <-> -. (-. ph \/ -. ps))))
6	1, 5	syld 40	1 |- (DECID ph -> (DECID ps -> ((ph /\ ps) <-> -. (-. ph \/ -. ps))))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 628  DECID wdc 741

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-in1 544  ax-in2 545  ax-io 629

This theorem depends on definitions:  df-bi 110  df-dc 742

This theorem is referenced by:  pm3.11dc  863  dn1dc  866