Theorem 3orbi123d 1311

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 793	. . 3 |- ( ph -> ( ( ps \/ th ) <-> ( ch \/ ta ) ) )
4		bi3d.3	. . 3 |- ( ph -> ( et <-> ze ) )
5	3, 4	orbi12d 793	. 2 |- ( ph -> ( ( ( ps \/ th ) \/ et ) <-> ( ( ch \/ ta ) \/ ze ) ) )
6		df-3or 979	. 2 |- ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
7		df-3or 979	. 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 708   \/ w3o 977

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 709

This theorem depends on definitions:  df-bi 117  df-3or 979

This theorem is referenced by:  ordtriexmid  4514  ontriexmidim  4515  wetriext  4570  nntri3or  6484  tridc  6889  exmidontriimlem3  7212  exmidontriimlem4  7213  exmidontriim  7214  onntri35  7226  ltsopi  7294  pitri3or  7296  nqtri3or  7370  elz  9228  ztri3or  9269  qtri3or  10213  trilpo  14352  trirec0  14353  reap0  14367