Theorem dfbi 573

Description: Definition df-bi 164 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.

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 572	. . 3 |- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
2	1	biimpi 168	. 2 |- ((ph <-> ps) -> ((ph -> ps) /\ (ps -> ph)))
3	1	biimpri 169	. 2 |- (((ph -> ps) /\ (ps -> ph)) -> (ph <-> ps))
4	2, 3	pm3.2i 307	1 |- (((ph <-> ps) -> ((ph -> ps) /\ (ps -> ph))) /\ (((ph -> ps) /\ (ps -> ph)) -> (ph <-> ps)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-an 242

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