Theorem imimorbdc 882

Description: Simplify an implication between implications, for a decidable proposition. (Contributed by Jim Kingdon, 18-Mar-2018.)

Assertion

Ref	Expression
imimorbdc	|- (DECID ps -> ( ( ( ps -> ch ) -> ( ph -> ch ) ) <-> ( ph -> ( ps \/ ch ) ) ) )

Proof of Theorem imimorbdc

Step	Hyp	Ref	Expression
1		bi2.04 247	. 2 |- ( ( ( ps -> ch ) -> ( ph -> ch ) ) <-> ( ph -> ( ( ps -> ch ) -> ch ) ) )
2		dfor2dc 881	. . 3 |- (DECID ps -> ( ( ps \/ ch ) <-> ( ( ps -> ch ) -> ch ) ) )
3	2	imbi2d 229	. 2 |- (DECID ps -> ( ( ph -> ( ps \/ ch ) ) <-> ( ph -> ( ( ps -> ch ) -> ch ) ) ) )
4	1, 3	bitr4id 198	1 |- (DECID ps -> ( ( ( ps -> ch ) -> ( ph -> ch ) ) <-> ( ph -> ( ps \/ ch ) ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 698  DECID wdc 820

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 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 821

This theorem is referenced by: (None)