Theorem dfordc 791

Description: Definition of 'or' in terms of negation and implication for a decidable proposition. Based on definition of [Margaris] p. 49. One direction, pm2.53 641, 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		pm2.53 641	. 2 |- ( ( ph \/ ps ) -> ( -. ph -> ps ) )
2		pm2.54dc 790	. 2 |- (DECID ph -> ( ( -. ph -> ps ) -> ( ph \/ ps ) ) )
3	1, 2	impbid2 131	1 |- (DECID ph -> ( ( ph \/ ps ) <-> ( -. ph -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   \/ wo 629  DECID wdc 742

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 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  imordc  796  pm4.64dc  801  pm5.17dc  810  pm5.6dc  835  pm3.12dc  865  pm5.15dc  1280  19.32dc  1569  r19.32vdc  2459  prime  8337