Theorem 3orbi123d 1289

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 782	. . 3 |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) )
4		bi3d.3	. . 3 |- ( ph -> ( et <-> ze ) )
5	3, 4	orbi12d 782	. 2 |- ( ph -> ( ( ( ps \/ th ) \/ et ) <-> ( ( ch \/ ta ) \/ ze ) ) )
6		df-3or 963	. 2 |- ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
7		df-3or 963	. 2 |- ( ( ch \/ ta \/ ze ) <-> ( ( ch \/ ta ) \/ ze ) )
8	5, 6, 7	3bitr4g 222	1 |- ( ph -> ( ( ps \/ th \/ et ) <-> ( ch \/ ta \/ ze ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 697   \/ w3o 961

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 698

This theorem depends on definitions:  df-bi 116  df-3or 963

This theorem is referenced by:  ordtriexmid  4432  wetriext  4486  nntri3or  6382  tridc  6786  ltsopi  7121  pitri3or  7123  nqtri3or  7197  elz  9049  ztri3or  9090  qtri3or  10013  trilpo  13225