Theorem 3orbi123i 1179

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 754	. . 3 |- ( ( ph \/ ch ) <-> ( ps \/ th ) )
4		bi3.3	. . 3 |- ( ta <-> et )
5	3, 4	orbi12i 754	. 2 |- ( ( ( ph \/ ch ) \/ ta ) <-> ( ( ps \/ th ) \/ et ) )
6		df-3or 969	. 2 |- ( ( ph \/ ch \/ ta ) <-> ( ( ph \/ ch ) \/ ta ) )
7		df-3or 969	. 2 |- ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
8	5, 6, 7	3bitr4i 211	1 |- ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) )

Syntax hints:   <-> wb 104   \/ wo 698   \/ w3o 967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699

This theorem depends on definitions:  df-bi 116  df-3or 969

This theorem is referenced by:  nnwetri  6881  exmidontriimlem3  7179