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 ) ) )