Theorem ibibr 245

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

Assertion

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

Proof of Theorem ibibr

Step	Hyp	Ref	Expression
1		pm5.501 243	. . 3 |- ( ph -> ( ps <-> ( ph <-> ps ) ) )
2		bicom 139	. . 3 |- ( ( ph <-> ps ) <-> ( ps <-> ph ) )
3	1, 2	bitrdi 195	. 2 |- ( ph -> ( ps <-> ( ps <-> ph ) ) )
4	3	pm5.74i 179	1 |- ( ( ph -> ps ) <-> ( ph -> ( ps <-> ph ) ) )

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:  tbt  246  oibabs  704  rabxfrd  4447