Theorem dfordc 825

Description: Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 674, 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		pm2.53 674	. 2 |- ( ( ph \/ ps ) -> ( -. ph -> ps ) )
2		pm2.54dc 824	. 2 |- (DECID ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) )
3	1, 2	impbid2 141	1 |- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   \/ wo 662  DECID wdc 776

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663

This theorem depends on definitions:  df-bi 115  df-dc 777

This theorem is referenced by:  imordc  830  pm4.64dc  835  pm5.17dc  844  pm5.6dc  869  pm3.12dc  900  pm5.15dc  1321  19.32dc  1610  r19.32vdc  2508  prime  8579  isprm4  10708  prm2orodd  10715  euclemma  10732  phiprmpw  10805