Theorem dfordc 893

Description: Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 723, 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		pm2.53 723	. 2 |- ( ( ph \/ ps ) -> ( -. ph -> ps ) )
2		pm2.54dc 892	. 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 709  DECID wdc 835

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 615  ax-in2 616  ax-io 710

This theorem depends on definitions:  df-bi 117  df-dc 836

This theorem is referenced by:  imordc  898  pm4.64dc  901  pm5.17dc  905  pm5.6dc  927  pm3.12dc  960  pm5.15dc  1400  19.32dc  1690  r19.30dc  2637  r19.32vdc  2639  prime  9370  isprm4  12137  prm2orodd  12144  euclemma  12164  phiprmpw  12240