Theorem annotanannot 676

Description: A conjunction with a negated conjunction. (Contributed by AV, 8-Mar-2022.) (Proof shortened by Wolf Lammen, 1-Apr-2022.)

Assertion

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

Proof of Theorem annotanannot

Step	Hyp	Ref	Expression
1		ibar 301	. . . 4 |- ( ph -> ( ps <-> ( ph /\ ps ) ) )
2	1	bicomd 141	. . 3 |- ( ph -> ( ( ph /\ ps ) <-> ps ) )
3	2	notbid 673	. 2 |- ( ph -> ( -. ( ph /\ ps ) <-> -. ps ) )
4	3	pm5.32i 454	1 |- ( ( ph /\ -. ( ph /\ ps ) ) <-> ( ph /\ -. ps ) )

Syntax hints:   -. wn 3   /\ wa 104   <-> 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 619  ax-in2 620

This theorem depends on definitions:  df-bi 117

This theorem is referenced by: (None)