Theorem pm4.87 356

Description: Theorem *4.87 of [WhiteheadRussell] p. 122. (The proof was 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 347	. . 3 |- (((ph /\ ps) -> ch) <-> (ph -> (ps -> ch)))
2		bi2.04 160	. . 3 |- ((ph -> (ps -> ch)) <-> (ps -> (ph -> ch)))
3	1, 2	pm3.2i 285	. 2 |- ((((ph /\ ps) -> ch) <-> (ph -> (ps -> ch))) /\ ((ph -> (ps -> ch)) <-> (ps -> (ph -> ch))))
4		impexp 347	. . 3 |- (((ps /\ ph) -> ch) <-> (ps -> (ph -> ch)))
5	4	bicomi 172	. 2 |- ((ps -> (ph -> ch)) <-> ((ps /\ ph) -> ch))
6	3, 5	pm3.2i 285	1 |- (((((ph /\ ps) -> ch) <-> (ph -> (ps -> ch))) /\ ((ph -> (ps -> ch)) <-> (ps -> (ph -> ch)))) /\ ((ps -> (ph -> ch)) <-> ((ps /\ ph) -> ch)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-an 225

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