Theorem 3orbi123d 1348

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 801	. . 3 |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) )
4		bi3d.3	. . 3 |- ( ph -> ( et <-> ze ) )
5	3, 4	orbi12d 801	. 2 |- ( ph -> ( ( ( ps \/ th ) \/ et ) <-> ( ( ch \/ ta ) \/ ze ) ) )
6		df-3or 1006	. 2 |- ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
7		df-3or 1006	. 2 |- ( ( ch \/ ta \/ ze ) <-> ( ( ch \/ ta ) \/ ze ) )
8	5, 6, 7	3bitr4g 223	1 |- ( ph -> ( ( ps \/ th \/ et ) <-> ( ch \/ ta \/ ze ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 716   \/ w3o 1004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717

This theorem depends on definitions:  df-bi 117  df-3or 1006

This theorem is referenced by:  ordtriexmid  4643  ontriexmidim  4644  wetriext  4699  nntri3or  6726  tridc  7157  exmidontriimlem3  7530  exmidontriimlem4  7531  exmidontriim  7532  onntri35  7547  ltsopi  7635  pitri3or  7637  nqtri3or  7711  elz  9579  ztri3or  9620  qtri3or  10600  trilpo  16827  trirec0  16828  reap0  16843