Theorem 3orbi123d 1345

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 798	. . 3 |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) )
4		bi3d.3	. . 3 |- ( ph -> ( et <-> ze ) )
5	3, 4	orbi12d 798	. 2 |- ( ph -> ( ( ( ps \/ th ) \/ et ) <-> ( ( ch \/ ta ) \/ ze ) ) )
6		df-3or 1003	. 2 |- ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
7		df-3or 1003	. 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 713   \/ w3o 1001

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 714

This theorem depends on definitions:  df-bi 117  df-3or 1003

This theorem is referenced by:  ordtriexmid  4613  ontriexmidim  4614  wetriext  4669  nntri3or  6639  tridc  7061  exmidontriimlem3  7405  exmidontriimlem4  7406  exmidontriim  7407  onntri35  7422  ltsopi  7507  pitri3or  7509  nqtri3or  7583  elz  9448  ztri3or  9489  qtri3or  10460  trilpo  16411  trirec0  16412  reap0  16426