Theorem notbi 667

Description: Equivalence property for negation. Closed form. (Contributed by BJ, 27-Jan-2020.)

Assertion

Ref	Expression
notbi	|- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )

Proof of Theorem notbi

Step	Hyp	Ref	Expression
1		biimpr 130	. . 3 |- ( ( ph <-> ps ) -> ( ps -> ph ) )
2	1	con3d 632	. 2 |- ( ( ph <-> ps ) -> ( -. ph -> -. ps ) )
3		biimp 118	. . 3 |- ( ( ph <-> ps ) -> ( ph -> ps ) )
4	3	con3d 632	. 2 |- ( ( ph <-> ps ) -> ( -. ps -> -. ph ) )
5	2, 4	impbid 129	1 |- ( ( ph <-> ps ) -> ( -. ph <-> -. ps ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  notbid  668  notbii  669  ifbi  3581