Theorem 3orbi123d 894

Description: Deduction joining 3 equivalences to form equivalence of disjunctions.

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 629	. . 3 |- (ph -> ((ps \/ th) <-> (ch \/ ta)))
4		bi3d.3	. . 3 |- (ph -> (et <-> ze))
5	3, 4	orbi12d 629	. 2 |- (ph -> (((ps \/ th) \/ et) <-> ((ch \/ ta) \/ ze)))
6		df-3or 778	. 2 |- ((ps \/ th \/ et) <-> ((ps \/ th) \/ et))
7		df-3or 778	. 2 |- ((ch \/ ta \/ ze) <-> ((ch \/ ta) \/ ze))
8	5, 6, 7	3bitr4g 557	1 |- (ph -> ((ps \/ th \/ et) <-> (ch \/ ta \/ ze)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   \/ w3o 776

This theorem is referenced by:  moeq3 1924  soeq1 2859  solin 2863  dfwe2 2941  weinxp 3239  isowe 3909  f1oweALT 3912  rdglem2 3944  ltsopr 5148  elz 6139

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778

Copyright terms: Public domain