Theorem imim21b 251

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.)

Assertion

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

Proof of Theorem imim21b

Step	Hyp	Ref	Expression
1		bi2.04 247	. 2 |- ( ( ( ph -> ch ) -> ( ps -> th ) ) <-> ( ps -> ( ( ph -> ch ) -> th ) ) )
2		pm5.5 241	. . . . 5 |- ( ph -> ( ( ph -> ch ) <-> ch ) )
3	2	imbi1d 230	. . . 4 |- ( ph -> ( ( ( ph -> ch ) -> th ) <-> ( ch -> th ) ) )
4	3	imim2i 12	. . 3 |- ( ( ps -> ph ) -> ( ps -> ( ( ( ph -> ch ) -> th ) <-> ( ch -> th ) ) ) )
5	4	pm5.74d 181	. 2 |- ( ( ps -> ph ) -> ( ( ps -> ( ( ph -> ch ) -> th ) ) <-> ( ps -> ( ch -> th ) ) ) )
6	1, 5	syl5bb 191	1 |- ( ( ps -> ph ) -> ( ( ( ph -> ch ) -> ( ps -> th ) ) <-> ( ps -> ( ch -> th ) ) ) )

Syntax hints:   -> wi 4   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)