Theorem ifpdfbidc 991

Description: Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020.) (Proof shortened by Wolf Lammen, 30-Apr-2024.)

Assertion

Ref	Expression
ifpdfbidc	|- (DECID ph -> ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) ) )

Proof of Theorem ifpdfbidc

Step	Hyp	Ref	Expression
1		con34bdc 876	. . 3 |- (DECID ph -> ( ( ps -> ph ) <-> ( -. ph -> -. ps ) ) )
2	1	anbi2d 464	. 2 |- (DECID ph -> ( ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> ( ( ph -> ps ) /\ ( -. ph -> -. ps ) ) ) )
3		dfbi2 388	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
4	3	a1i 9	. 2 |- (DECID ph -> ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) )
5		dfifp2dc 987	. 2 |- (DECID ph -> (if- ( ph , ps , -. ps ) <-> ( ( ph -> ps ) /\ ( -. ph -> -. ps ) ) ) )
6	2, 4, 5	3bitr4d 220	1 |- (DECID ph -> ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105  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-stab 836  df-dc 840  df-ifp 984

This theorem is referenced by: (None)