Theorem ifpbi123dOLD

Description: Obsolete version of ifpbi123d as of 17-Apr-2024. (Contributed by AV, 30-Dec-2020) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	ifpbi123d.1	|- ( ph -> ( ps <-> ta ) )
	ifpbi123d.2	|- ( ph -> ( ch <-> et ) )
	ifpbi123d.3	|- ( ph -> ( th <-> ze ) )
Assertion	ifpbi123dOLD	|- ( ph -> ( if- ( ps , ch , th ) <-> if- ( ta , et , ze ) ) )

Proof

Step	Hyp	Ref	Expression
1		ifpbi123d.1	|- ( ph -> ( ps <-> ta ) )
2		ifpbi123d.2	|- ( ph -> ( ch <-> et ) )
3		ifpbi123d.3	|- ( ph -> ( th <-> ze ) )
4	1 2	anbi12d	|- ( ph -> ( ( ps /\ ch ) <-> ( ta /\ et ) ) )
5	1	notbid	|- ( ph -> ( -. ps <-> -. ta ) )
6	5 3	anbi12d	|- ( ph -> ( ( -. ps /\ th ) <-> ( -. ta /\ ze ) ) )
7	4 6	orbi12d	|- ( ph -> ( ( ( ps /\ ch ) \/ ( -. ps /\ th ) ) <-> ( ( ta /\ et ) \/ ( -. ta /\ ze ) ) ) )
8		df-ifp	|- ( if- ( ps , ch , th ) <-> ( ( ps /\ ch ) \/ ( -. ps /\ th ) ) )
9		df-ifp	|- ( if- ( ta , et , ze ) <-> ( ( ta /\ et ) \/ ( -. ta /\ ze ) ) )
10	7 8 9	3bitr4g	|- ( ph -> ( if- ( ps , ch , th ) <-> if- ( ta , et , ze ) ) )