Metamath Proof Explorer


Theorem ibibr

Description: Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005) (Proof shortened by Wolf Lammen, 21-Dec-2013)

Ref Expression
Assertion ibibr
|- ( ( ph -> ps ) <-> ( ph -> ( ps <-> ph ) ) )

Proof

Step Hyp Ref Expression
1 pm5.501
 |-  ( ph -> ( ps <-> ( ph <-> ps ) ) )
2 bicom
 |-  ( ( ph <-> ps ) <-> ( ps <-> ph ) )
3 1 2 bitrdi
 |-  ( ph -> ( ps <-> ( ps <-> ph ) ) )
4 3 pm5.74i
 |-  ( ( ph -> ps ) <-> ( ph -> ( ps <-> ph ) ) )