Theorem 3orbi123i 1252

Description: Join 3 biconditionals with disjunction. (Contributed by NM, 17-May-1994.)

Hypotheses

Ref	Expression
bi3.1	|- ( ph <-> ps )
bi3.2	|- ( ch <-> th )
bi3.3	|- ( ta <-> et )

Assertion

Ref	Expression
3orbi123i	|- ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) )

Proof of Theorem 3orbi123i

Step	Hyp	Ref	Expression
1		bi3.1	. . . 4 |- ( ph <-> ps )
2		bi3.2	. . . 4 |- ( ch <-> th )
3	1, 2	orbi12i 543	. . 3 |- ( ( ph \/ ch ) <-> ( ps \/ th ) )
4		bi3.3	. . 3 |- ( ta <-> et )
5	3, 4	orbi12i 543	. 2 |- ( ( ( ph \/ ch ) \/ ta ) <-> ( ( ps \/ th ) \/ et ) )
6		df-3or 1038	. 2 |- ( ( ph \/ ch \/ ta ) <-> ( ( ph \/ ch ) \/ ta ) )
7		df-3or 1038	. 2 |- ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
8	5, 6, 7	3bitr4i 292	1 |- ( ( ph \/ ch \/ ta ) <-> ( ps \/ th \/ et ) )

Syntax hints:   <-> wb 196   \/ wo 383   \/ w3o 1036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-3or 1038

This theorem is referenced by:  ne3anior  2887  wecmpep  5106  cnvso  5674  sorpss  6942  ordon  6982  soxp  7290  dford2  8517  elfz0lmr  12583  axlowdimlem6  25827  elxrge02  29640  brtp  31639  dfon2  31697  sltsolem1  31826  frege129d  38055  dfxlim2  40074