Theorem nan 693

Description: Theorem to move a conjunct in and out of a negation. (Contributed by NM, 9-Nov-2003.)

Assertion

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

Proof of Theorem nan

Step	Hyp	Ref	Expression
1		impexp 263	. 2 |- ( ( ( ph /\ ps ) -> -. ch ) <-> ( ph -> ( ps -> -. ch ) ) )
2		imnan 691	. . 3 |- ( ( ps -> -. ch ) <-> -. ( ps /\ ch ) )
3	2	imbi2i 226	. 2 |- ( ( ph -> ( ps -> -. ch ) ) <-> ( ph -> -. ( ps /\ ch ) ) )
4	1, 3	bitr2i 185	1 |- ( ( ph -> -. ( ps /\ ch ) ) <-> ( ( ph /\ ps ) -> -. ch ) )

Syntax hints:   -. wn 3   -> wi 4   /\ 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 615  ax-in2 616

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  pm4.15  695