Theorem dfordc 887

Description: Definition of disjunction in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 717, 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		pm2.53 717	. 2 |- ( ( ph \/ ps ) -> ( -. ph -> ps ) )
2		pm2.54dc 886	. 2 |- (DECID ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) )
3	1, 2	impbid2 142	1 |- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   \/ wo 703  DECID wdc 829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-dc 830

This theorem is referenced by:  imordc  892  pm4.64dc  895  pm5.17dc  899  pm5.6dc  921  pm3.12dc  953  pm5.15dc  1384  19.32dc  1672  r19.30dc  2617  r19.32vdc  2619  prime  9311  isprm4  12073  prm2orodd  12080  euclemma  12100  phiprmpw  12176