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Metamath Proof Explorer

Theorem 3or6

Description: Analogue of or4 for triple conjunction. (Contributed by Scott Fenton, 16-Mar-2011)

Ref Expression
Assertion 3or6 
|- ( ( ( ph \/ ps ) \/ ( ch \/ th ) \/ ( ta \/ et ) ) <-> ( ( ph \/ ch \/ ta ) \/ ( ps \/ th \/ et ) ) )

Proof

Step Hyp Ref Expression
1 or4 
 |-  ( ( ( ( ph \/ ch ) \/ ta ) \/ ( ( ps \/ th ) \/ et ) ) <-> ( ( ( ph \/ ch ) \/ ( ps \/ th ) ) \/ ( ta \/ et ) ) )
2 or4 
 |-  ( ( ( ph \/ ch ) \/ ( ps \/ th ) ) <-> ( ( ph \/ ps ) \/ ( ch \/ th ) ) )
3 2 orbi1i 
 |-  ( ( ( ( ph \/ ch ) \/ ( ps \/ th ) ) \/ ( ta \/ et ) ) <-> ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) \/ ( ta \/ et ) ) )
4 1 3 bitr2i 
 |-  ( ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) \/ ( ta \/ et ) ) <-> ( ( ( ph \/ ch ) \/ ta ) \/ ( ( ps \/ th ) \/ et ) ) )
5 df-3or 
 |-  ( ( ( ph \/ ps ) \/ ( ch \/ th ) \/ ( ta \/ et ) ) <-> ( ( ( ph \/ ps ) \/ ( ch \/ th ) ) \/ ( ta \/ et ) ) )
6 df-3or 
 |-  ( ( ph \/ ch \/ ta ) <-> ( ( ph \/ ch ) \/ ta ) )
7 df-3or 
 |-  ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
8 6 7 orbi12i 
 |-  ( ( ( ph \/ ch \/ ta ) \/ ( ps \/ th \/ et ) ) <-> ( ( ( ph \/ ch ) \/ ta ) \/ ( ( ps \/ th ) \/ et ) ) )
9 4 5 8 3bitr4i 
 |-  ( ( ( ph \/ ps ) \/ ( ch \/ th ) \/ ( ta \/ et ) ) <-> ( ( ph \/ ch \/ ta ) \/ ( ps \/ th \/ et ) ) )