Theorem dfbi 610

Description: Definition df-bi 177 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 609	. . 3 |- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
2	1	biimpi 186	. 2 |- ((ph <-> ps) -> ((ph -> ps) /\ (ps -> ph)))
3	1	biimpri 197	. 2 |- (((ph -> ps) /\ (ps -> ph)) -> (ph <-> ps))
4	2, 3	pm3.2i 441	1 |- (((ph <-> ps) -> ((ph -> ps) /\ (ps -> ph))) /\ (((ph -> ps) /\ (ps -> ph)) -> (ph <-> ps)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358

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 177  df-an 360

This theorem is referenced by: (None)