Theorem pm5.54 943

Description: Theorem *5.54 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 7-Nov-2013.)

Assertion

Ref	Expression
pm5.54	|- ( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) )

Proof of Theorem pm5.54

Step	Hyp	Ref	Expression
1		iba 524	. . . . 5 |- ( ps -> ( ph <-> ( ph /\ ps ) ) )
2	1	bicomd 213	. . . 4 |- ( ps -> ( ( ph /\ ps ) <-> ph ) )
3	2	adantl 482	. . 3 |- ( ( ph /\ ps ) -> ( ( ph /\ ps ) <-> ph ) )
4	3, 2	pm5.21ni 367	. 2 |- ( -. ( ( ph /\ ps ) <-> ph ) -> ( ( ph /\ ps ) <-> ps ) )
5	4	orri 391	1 |- ( ( ( ph /\ ps ) <-> ph ) \/ ( ( ph /\ ps ) <-> ps ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384

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 197  df-or 385  df-an 386

This theorem is referenced by: (None)