Theorem pm4.87 567

Description: Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)

Assertion

Ref	Expression
pm4.87	|- (((((ph /\ ps) -> ch) <-> (ph -> (ps -> ch))) /\ ((ph -> (ps -> ch)) <-> (ps -> (ph -> ch)))) /\ ((ps -> (ph -> ch)) <-> ((ps /\ ph) -> ch)))

Proof of Theorem pm4.87

Step	Hyp	Ref	Expression
1		impexp 433	. . 3 |- (((ph /\ ps) -> ch) <-> (ph -> (ps -> ch)))
2		bi2.04 350	. . 3 |- ((ph -> (ps -> ch)) <-> (ps -> (ph -> ch)))
3	1, 2	pm3.2i 441	. 2 |- ((((ph /\ ps) -> ch) <-> (ph -> (ps -> ch))) /\ ((ph -> (ps -> ch)) <-> (ps -> (ph -> ch))))
4		impexp 433	. . 3 |- (((ps /\ ph) -> ch) <-> (ps -> (ph -> ch)))
5	4	bicomi 193	. 2 |- ((ps -> (ph -> ch)) <-> ((ps /\ ph) -> ch))
6	3, 5	pm3.2i 441	1 |- (((((ph /\ ps) -> ch) <-> (ph -> (ps -> ch))) /\ ((ph -> (ps -> ch)) <-> (ps -> (ph -> ch)))) /\ ((ps -> (ph -> ch)) <-> ((ps /\ ph) -> ch)))

Syntax hints:   -> 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: (None)