Theorem 3orbi123i 1141

Description: Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)

Hypotheses

Ref	Expression
bi3.1	|- (ph <-> ps)
bi3.2	|- (ch <-> th)
bi3.3	|- (ta <-> et)

Assertion

Ref	Expression
3orbi123i	|- ((ph \/ ch \/ ta) <-> (ps \/ th \/ et))

Proof of Theorem 3orbi123i

Step	Hyp	Ref	Expression
1		bi3.1	. . . 4 |- (ph <-> ps)
2		bi3.2	. . . 4 |- (ch <-> th)
3	1, 2	orbi12i 507	. . 3 |- ((ph \/ ch) <-> (ps \/ th))
4		bi3.3	. . 3 |- (ta <-> et)
5	3, 4	orbi12i 507	. 2 |- (((ph \/ ch) \/ ta) <-> ((ps \/ th) \/ et))
6		df-3or 935	. 2 |- ((ph \/ ch \/ ta) <-> ((ph \/ ch) \/ ta))
7		df-3or 935	. 2 |- ((ps \/ th \/ et) <-> ((ps \/ th) \/ et))
8	5, 6, 7	3bitr4i 268	1 |- ((ph \/ ch \/ ta) <-> (ps \/ th \/ et))

Syntax hints:   <-> 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:  cadcomb  1396  ne3anior  2602