Theorem dfbi 661

Description: Definition df-bi 197 rewritten in an abbreviated form to help intuitive understanding of that definition. Note that it is a conjunction of two implications; one which asserts properties that follow from the biconditional and one which asserts properties that imply the biconditional. (Contributed by NM, 15-Aug-2008.)

Assertion

Ref	Expression
dfbi	|- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) )

Proof of Theorem dfbi

Step	Hyp	Ref	Expression
1		dfbi2 660	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	biimpi 206	. 2 |- ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3	1	biimpri 218	. 2 |- ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) )
4	2, 3	pm3.2i 471	1 |- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-an 386

This theorem is referenced by: (None)