Theorem ifpbi123d

Description: Equivalence deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020) (Proof shortened by Wolf Lammen, 17-Apr-2024)

	Ref	Expression
Hypotheses	ifpbi123d.1	|- ( ph -> ( ps <-> ta ) )
	ifpbi123d.2	|- ( ph -> ( ch <-> et ) )
	ifpbi123d.3	|- ( ph -> ( th <-> ze ) )
Assertion	ifpbi123d	|- ( 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	imbi12d	|- ( ph -> ( ( ps -> ch ) <-> ( ta -> et ) ) )
5	1 3	orbi12d	|- ( ph -> ( ( ps \/ th ) <-> ( ta \/ ze ) ) )
6	4 5	anbi12d	|- ( ph -> ( ( ( ps -> ch ) /\ ( ps \/ th ) ) <-> ( ( ta -> et ) /\ ( ta \/ ze ) ) ) )
7		dfifp3	|- ( if- ( ps , ch , th ) <-> ( ( ps -> ch ) /\ ( ps \/ th ) ) )
8		dfifp3	|- ( if- ( ta , et , ze ) <-> ( ( ta -> et ) /\ ( ta \/ ze ) ) )
9	6 7 8	3bitr4g	|- ( ph -> ( if- ( ps , ch , th ) <-> if- ( ta , et , ze ) ) )