Theorem dfifp4dc 989

Description: Alternate definition of the conditional operator for propositions. (Contributed by BJ, 30-Sep-2019.)

Assertion

Ref	Expression
dfifp4dc	|- (DECID ph -> (if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) ) )

Proof of Theorem dfifp4dc

Step	Hyp	Ref	Expression
1		dfifp3dc 988	. 2 |- (DECID ph -> (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) ) )
2		imordc 902	. . 3 |- (DECID ph -> ( ( ph -> ps ) <-> ( -. ph \/ ps ) ) )
3	2	anbi1d 465	. 2 |- (DECID ph -> ( ( ( ph -> ps ) /\ ( ph \/ ch ) ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) ) )
4	1, 3	bitrd 188	1 |- (DECID ph -> (if- ( ph , ps , ch ) <-> ( ( -. ph \/ ps ) /\ ( ph \/ ch ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839  if-wif 983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984

This theorem is referenced by:  anifpdc  992