Theorem nan 563

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 433	. 2 |- (((ph /\ ps) -> -. ch) <-> (ph -> (ps -> -. ch)))
2		imnan 411	. . 3 |- ((ps -> -. ch) <-> -. (ps /\ ch))
3	2	imbi2i 303	. 2 |- ((ph -> (ps -> -. ch)) <-> (ph -> -. (ps /\ ch)))
4	1, 3	bitr2i 241	1 |- ((ph -> -. (ps /\ ch)) <-> ((ph /\ ps) -> -. ch))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358

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 177  df-an 360

This theorem is referenced by:  pm4.15  564