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Metamath Proof Explorer

Theorem pm4.78

Description: Implication distributes over disjunction. Theorem *4.78 of WhiteheadRussell p. 121. (Contributed by NM, 3-Jan-2005) (Proof shortened by Wolf Lammen, 19-Nov-2012)

Ref Expression
Assertion pm4.78 
|- ( ( ( ph -> ps ) \/ ( ph -> ch ) ) <-> ( ph -> ( ps \/ ch ) ) )

Proof

Step Hyp Ref Expression
1 orordi 
 |-  ( ( -. ph \/ ( ps \/ ch ) ) <-> ( ( -. ph \/ ps ) \/ ( -. ph \/ ch ) ) )
2 imor 
 |-  ( ( ph -> ( ps \/ ch ) ) <-> ( -. ph \/ ( ps \/ ch ) ) )
3 imor 
 |-  ( ( ph -> ps ) <-> ( -. ph \/ ps ) )
4 imor 
 |-  ( ( ph -> ch ) <-> ( -. ph \/ ch ) )
5 3 4 orbi12i 
 |-  ( ( ( ph -> ps ) \/ ( ph -> ch ) ) <-> ( ( -. ph \/ ps ) \/ ( -. ph \/ ch ) ) )
6 1 2 5 3bitr4ri 
 |-  ( ( ( ph -> ps ) \/ ( ph -> ch ) ) <-> ( ph -> ( ps \/ ch ) ) )