Theorem 3orbi123d 1251

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

Hypotheses

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

Assertion

Ref	Expression
3orbi123d	|- (ph -> ((ps \/ th \/ et) <-> (ch \/ ta \/ ze)))

Proof of Theorem 3orbi123d

Step	Hyp	Ref	Expression
1		bi3d.1	. . . 4 |- (ph -> (ps <-> ch))
2		bi3d.2	. . . 4 |- (ph -> (th <-> ta))
3	1, 2	orbi12d 690	. . 3 |- (ph -> ((ps \/ th) <-> (ch \/ ta)))
4		bi3d.3	. . 3 |- (ph -> (et <-> ze))
5	3, 4	orbi12d 690	. 2 |- (ph -> (((ps \/ th) \/ et) <-> ((ch \/ ta) \/ ze)))
6		df-3or 935	. 2 |- ((ps \/ th \/ et) <-> ((ps \/ th) \/ et))
7		df-3or 935	. 2 |- ((ch \/ ta \/ ze) <-> ((ch \/ ta) \/ ze))
8	5, 6, 7	3bitr4g 279	1 |- (ph -> ((ps \/ th \/ et) <-> (ch \/ ta \/ ze)))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   \/ w3o 933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359  df-3or 935

This theorem is referenced by:  moeq3  3014  ltfintri  4467  nncdiv3  6278