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Metamath Proof Explorer

Theorem pm4.14

Description: Theorem *4.14 of WhiteheadRussell p. 117. Related to con34b . (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 23-Oct-2012)

Ref Expression
Assertion pm4.14 
|- ( ( ( ph /\ ps ) -> ch ) <-> ( ( ph /\ -. ch ) -> -. ps ) )

Proof

Step Hyp Ref Expression
1 con34b 
 |-  ( ( ps -> ch ) <-> ( -. ch -> -. ps ) )
2 1 imbi2i 
 |-  ( ( ph -> ( ps -> ch ) ) <-> ( ph -> ( -. ch -> -. ps ) ) )
3 impexp 
 |-  ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) )
4 impexp 
 |-  ( ( ( ph /\ -. ch ) -> -. ps ) <-> ( ph -> ( -. ch -> -. ps ) ) )
5 2 3 4 3bitr4i 
 |-  ( ( ( ph /\ ps ) -> ch ) <-> ( ( ph /\ -. ch ) -> -. ps ) )