Theorem 3orbi123d

Description: Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994)

	Ref	Expression
Hypotheses	bi3d.1	|- ( ph -> ( ps <-> ch ) )
	bi3d.2	|- ( ph -> ( th <-> ta ) )
	bi3d.3	|- ( ph -> ( et <-> ze ) )
Assertion	3orbi123d	|- ( ph -> ( ( ps \/ th \/ et ) <-> ( ch \/ ta \/ ze ) ) )

Proof

Step	Hyp	Ref	Expression
1		bi3d.1	|- ( ph -> ( ps <-> ch ) )
2		bi3d.2	|- ( ph -> ( th <-> ta ) )
3		bi3d.3	|- ( ph -> ( et <-> ze ) )
4	1 2	orbi12d	|- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) )
5	4 3	orbi12d	|- ( ph -> ( ( ( ps \/ th ) \/ et ) <-> ( ( ch \/ ta ) \/ ze ) ) )
6		df-3or	|- ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
7		df-3or	|- ( ( ch \/ ta \/ ze ) <-> ( ( ch \/ ta ) \/ ze ) )
8	5 6 7	3bitr4g	|- ( ph -> ( ( ps \/ th \/ et ) <-> ( ch \/ ta \/ ze ) ) )