Theorem pm4.87 547

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 261	. . 3 |- ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) )
2		bi2.04 247	. . 3 |- ( ( ph -> ( ps -> ch ) ) <-> ( ps -> ( ph -> ch ) ) )
3	1, 2	pm3.2i 270	. 2 |- ( ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) /\ ( ( ph -> ( ps -> ch ) ) <-> ( ps -> ( ph -> ch ) ) ) )
4		impexp 261	. . 3 |- ( ( ( ps /\ ph ) -> ch ) <-> ( ps -> ( ph -> ch ) ) )
5	4	bicomi 131	. 2 |- ( ( ps -> ( ph -> ch ) ) <-> ( ( ps /\ ph ) -> ch ) )
6	3, 5	pm3.2i 270	1 |- ( ( ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) ) /\ ( ( ph -> ( ps -> ch ) ) <-> ( ps -> ( ph -> ch ) ) ) ) /\ ( ( ps -> ( ph -> ch ) ) <-> ( ( ps /\ ph ) -> ch ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)