Theorem con34bdc 883

Description: Contraposition. Theorem *4.1 of [WhiteheadRussell] p. 116, but for a decidable proposition. (Contributed by Jim Kingdon, 24-Apr-2018.)

Assertion

Ref	Expression
con34bdc	|- (DECID ps -> ( ( ph -> ps ) <-> ( -. ps -> -. ph ) ) )

Proof of Theorem con34bdc

Step	Hyp	Ref	Expression
1		con3 651	. 2 |- ( ( ph -> ps ) -> ( -. ps -> -. ph ) )
2		condc 865	. 2 |- (DECID ps -> ( ( -. ps -> -. ph ) -> ( ph -> ps ) ) )
3	1, 2	impbid2 143	1 |- (DECID ps -> ( ( ph -> ps ) <-> ( -. ps -> -. ph ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   <-> wb 105  DECID wdc 846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847

This theorem is used by:  pm4.14dc  902  ifpdfbidc  998  algcvgblem  12827