Theorem ifpdfbi

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

	Ref	Expression
Assertion	ifpdfbi	|- ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) )

Proof

Step	Hyp	Ref	Expression
1		con34b	|- ( ( ps -> ph ) <-> ( -. ph -> -. ps ) )
2	1	anbi2i	|- ( ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> ( ( ph -> ps ) /\ ( -. ph -> -. ps ) ) )
3		dfbi2	|- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
4		dfifp2	|- ( if- ( ph , ps , -. ps ) <-> ( ( ph -> ps ) /\ ( -. ph -> -. ps ) ) )
5	2 3 4	3bitr4i	|- ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) )