Theorem imim21b

Description: Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011) (Proof shortened by Wolf Lammen, 14-Sep-2013)

	Ref	Expression
Assertion	imim21b	|- ( ( ps -> ph ) -> ( ( ( ph -> ch ) -> ( ps -> th ) ) <-> ( ps -> ( ch -> th ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		bi2.04	|- ( ( ( ph -> ch ) -> ( ps -> th ) ) <-> ( ps -> ( ( ph -> ch ) -> th ) ) )
2		pm5.5	|- ( ph -> ( ( ph -> ch ) <-> ch ) )
3	2	imbi1d	|- ( ph -> ( ( ( ph -> ch ) -> th ) <-> ( ch -> th ) ) )
4	3	imim2i	|- ( ( ps -> ph ) -> ( ps -> ( ( ( ph -> ch ) -> th ) <-> ( ch -> th ) ) ) )
5	4	pm5.74d	|- ( ( ps -> ph ) -> ( ( ps -> ( ( ph -> ch ) -> th ) ) <-> ( ps -> ( ch -> th ) ) ) )
6	1 5	syl5bb	|- ( ( ps -> ph ) -> ( ( ( ph -> ch ) -> ( ps -> th ) ) <-> ( ps -> ( ch -> th ) ) ) )