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Metamath Proof Explorer

Theorem sbc3or

Description: sbcor with a 3-disjuncts. This proof is sbc3orgVD automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011) (Revised by NM, 24-Aug-2018) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion sbc3or 
|- ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) )

Proof

Step Hyp Ref Expression
1 sbcor 
 |-  ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) )
2 df-3or 
 |-  ( ( ph \/ ps \/ ch ) <-> ( ( ph \/ ps ) \/ ch ) )
3 2 bicomi 
 |-  ( ( ( ph \/ ps ) \/ ch ) <-> ( ph \/ ps \/ ch ) )
4 3 sbcbii 
 |-  ( [. A / x ]. ( ( ph \/ ps ) \/ ch ) <-> [. A / x ]. ( ph \/ ps \/ ch ) )
5 sbcor 
 |-  ( [. A / x ]. ( ph \/ ps ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps ) )
6 5 orbi1i 
 |-  ( ( [. A / x ]. ( ph \/ ps ) \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) )
7 1 4 6 3bitr3i 
 |-  ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) )
8 df-3or 
 |-  ( ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) <-> ( ( [. A / x ]. ph \/ [. A / x ]. ps ) \/ [. A / x ]. ch ) )
9 7 8 bitr4i 
 |-  ( [. A / x ]. ( ph \/ ps \/ ch ) <-> ( [. A / x ]. ph \/ [. A / x ]. ps \/ [. A / x ]. ch ) )