Theorem ibibr 244

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		pm5.501 242	. . 3 |- ( ph -> ( ps <-> ( ph <-> ps ) ) )
2		bicom 138	. . 3 |- ( ( ph <-> ps ) <-> ( ps <-> ph ) )
3	1, 2	syl6bb 194	. 2 |- ( ph -> ( ps <-> ( ps <-> ph ) ) )
4	3	pm5.74i 178	1 |- ( ( ph -> ps ) <-> ( ph -> ( ps <-> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 103

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  tbt  245  oibabs  833  rabxfrd  4219