Theorem bj-dfbi6 32560

Description: Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.)

Assertion

Ref	Expression
bj-dfbi6	|- ( ( ph <-> ps ) <-> ( ( ph \/ ps ) <-> ( ph /\ ps ) ) )

Proof of Theorem bj-dfbi6

Step	Hyp	Ref	Expression
1		bj-dfbi5 32559	. 2 |- ( ( ph <-> ps ) <-> ( ( ph \/ ps ) -> ( ph /\ ps ) ) )
2		id 22	. . . 4 |- ( ( ( ph \/ ps ) -> ( ph /\ ps ) ) -> ( ( ph \/ ps ) -> ( ph /\ ps ) ) )
3		animorr 506	. . . 4 |- ( ( ph /\ ps ) -> ( ph \/ ps ) )
4	2, 3	impbid1 215	. . 3 |- ( ( ( ph \/ ps ) -> ( ph /\ ps ) ) -> ( ( ph \/ ps ) <-> ( ph /\ ps ) ) )
5		biimp 205	. . 3 |- ( ( ( ph \/ ps ) <-> ( ph /\ ps ) ) -> ( ( ph \/ ps ) -> ( ph /\ ps ) ) )
6	4, 5	impbii 199	. 2 |- ( ( ( ph \/ ps ) -> ( ph /\ ps ) ) <-> ( ( ph \/ ps ) <-> ( ph /\ ps ) ) )
7	1, 6	bitri 264	1 |- ( ( ph <-> ps ) <-> ( ( ph \/ ps ) <-> ( ph /\ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ 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-or 385  df-an 386

This theorem is referenced by: (None)