Theorem pm4.14 561

Description: Theorem *4.14 of [WhiteheadRussell] p. 117. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 23-Oct-2012.)

Assertion

Ref	Expression
pm4.14	|- (((ph /\ ps) -> ch) <-> ((ph /\ -. ch) -> -. ps))

Proof of Theorem pm4.14

Step	Hyp	Ref	Expression
1		con34b 283	. . 3 |- ((ps -> ch) <-> (-. ch -> -. ps))
2	1	imbi2i 303	. 2 |- ((ph -> (ps -> ch)) <-> (ph -> (-. ch -> -. ps)))
3		impexp 433	. 2 |- (((ph /\ ps) -> ch) <-> (ph -> (ps -> ch)))
4		impexp 433	. 2 |- (((ph /\ -. ch) -> -. ps) <-> (ph -> (-. ch -> -. ps)))
5	2, 3, 4	3bitr4i 268	1 |- (((ph /\ ps) -> ch) <-> ((ph /\ -. ch) -> -. ps))

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:  pm3.37  562