Theorem dfordc 897

Description: Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 727, holds for all propositions, not just decidable ones. (Contributed by Jim Kingdon, 26-Mar-2018.)

Assertion

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

Proof of Theorem dfordc

Step	Hyp	Ref	Expression
1		pm2.53 727	. 2 |- ( ( ph \/ ps ) -> ( -. ph -> ps ) )
2		pm2.54dc 896	. 2 |- (DECID ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) )
3	1, 2	impbid2 143	1 |- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 713  DECID wdc 839

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 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-dc 840

This theorem is referenced by:  imordc  902  pm4.64dc  905  pm5.17dc  909  pm5.6dc  931  pm3.12dc  964  pm5.15dc  1431  19.32dc  1725  r19.30dc  2678  r19.32vdc  2680  prime  9546  isprm4  12641  prm2orodd  12648  euclemma  12668  phiprmpw  12744