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Metamath Proof Explorer

Theorem pm4.72

Description: Implication in terms of biconditional and disjunction. Theorem *4.72 of WhiteheadRussell p. 121. (Contributed by NM, 30-Aug-1993) (Proof shortened by Wolf Lammen, 30-Jan-2013)

Ref Expression
Assertion pm4.72 
|- ( ( ph -> ps ) <-> ( ps <-> ( ph \/ ps ) ) )

Proof

Step Hyp Ref Expression
1 olc 
 |-  ( ps -> ( ph \/ ps ) )
2 pm2.621 
 |-  ( ( ph -> ps ) -> ( ( ph \/ ps ) -> ps ) )
3 1 2 impbid2 
 |-  ( ( ph -> ps ) -> ( ps <-> ( ph \/ ps ) ) )
4 orc 
 |-  ( ph -> ( ph \/ ps ) )
5 biimpr 
 |-  ( ( ps <-> ( ph \/ ps ) ) -> ( ( ph \/ ps ) -> ps ) )
6 4 5 syl5 
 |-  ( ( ps <-> ( ph \/ ps ) ) -> ( ph -> ps ) )
7 3 6 impbii 
 |-  ( ( ph -> ps ) <-> ( ps <-> ( ph \/ ps ) ) )