Theorem dfifp3dc 988

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

Assertion

Ref	Expression
dfifp3dc	|- (DECID ph -> (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( ph \/ ch ) ) ) )

Proof of Theorem dfifp3dc

Step	Hyp	Ref	Expression
1		dfifp2dc 987	. 2 |- (DECID ph -> (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) )
2		pm4.64dc 905	. . 3 |- (DECID ph -> ( ( -. ph -> ch ) <-> ( ph \/ ch ) ) )
3	2	anbi2d 464	. 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:  dfifp4dc  989  ifpnst  994