Theorem 3orbi123d 1322

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 794	. . 3 |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) )
4		bi3d.3	. . 3 |- ( ph -> ( et <-> ze ) )
5	3, 4	orbi12d 794	. 2 |- ( ph -> ( ( ( ps \/ th ) \/ et ) <-> ( ( ch \/ ta ) \/ ze ) ) )
6		df-3or 981	. 2 |- ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
7		df-3or 981	. 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 709   \/ w3o 979

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 710

This theorem depends on definitions:  df-bi 117  df-3or 981

This theorem is referenced by:  ordtriexmid  4557  ontriexmidim  4558  wetriext  4613  nntri3or  6551  tridc  6960  exmidontriimlem3  7290  exmidontriimlem4  7291  exmidontriim  7292  onntri35  7304  ltsopi  7387  pitri3or  7389  nqtri3or  7463  elz  9328  ztri3or  9369  qtri3or  10330  trilpo  15687  trirec0  15688  reap0  15702