Theorem ifpdfbi

Description: Define the biconditional as conditional logic operator. (Contributed by RP, 20-Apr-2020) (Proof shortened by Wolf Lammen, 30-Apr-2024) (Proof shortened by Garrett Katz, 25-Jun-2026)

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

Proof

Step	Hyp	Ref	Expression
1		dfbi3	|- ( ( ph <-> ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) )
2		df-ifp	|- ( if- ( ph , ps , -. ps ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ -. ps ) ) )
3	1 2	bitr4i	|- ( ( ph <-> ps ) <-> if- ( ph , ps , -. ps ) )