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Metamath Proof Explorer

Theorem pm5.54

Description: Theorem *5.54 of WhiteheadRussell p. 125. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 7-Nov-2013)

Ref Expression
Assertion pm5.54 
|- ( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) )

Proof

Step Hyp Ref Expression
1 iba 
 |-  ( ps -> ( ph <-> ( ph /\ ps ) ) )
2 1 bicomd 
 |-  ( ps -> ( ( ph /\ ps ) <-> ph ) )
3 2 adantl 
 |-  ( ( ph /\ ps ) -> ( ( ph /\ ps ) <-> ph ) )
4 3 2 pm5.21ni 
 |-  ( -. ( ( ph /\ ps ) <-> ph ) -> ( ( ph /\ ps ) <-> ps ) )
5 4 orri 
 |-  ( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) )