Theorem biluk 974

Description: Lukasiewicz's shortest axiom for equivalential calculus. Storrs McCall, ed., Polish Logic 1920-1939 (Oxford, 1967), p. 96. (Contributed by NM, 10-Jan-2005.)

Assertion

Ref	Expression
biluk	|- ( ( ph <-> ps ) <-> ( ( ch <-> ps ) <-> ( ph <-> ch ) ) )

Proof of Theorem biluk

Step	Hyp	Ref	Expression
1		bicom 212	. . . . 5 |- ( ( ph <-> ps ) <-> ( ps <-> ph ) )
2	1	bibi1i 328	. . . 4 |- ( ( ( ph <-> ps ) <-> ch ) <-> ( ( ps <-> ph ) <-> ch ) )
3		biass 374	. . . 4 |- ( ( ( ps <-> ph ) <-> ch ) <-> ( ps <-> ( ph <-> ch ) ) )
4	2, 3	bitri 264	. . 3 |- ( ( ( ph <-> ps ) <-> ch ) <-> ( ps <-> ( ph <-> ch ) ) )
5		biass 374	. . 3 |- ( ( ( ( ph <-> ps ) <-> ch ) <-> ( ps <-> ( ph <-> ch ) ) ) <-> ( ( ph <-> ps ) <-> ( ch <-> ( ps <-> ( ph <-> ch ) ) ) ) )
6	4, 5	mpbi 220	. 2 |- ( ( ph <-> ps ) <-> ( ch <-> ( ps <-> ( ph <-> ch ) ) ) )
7		biass 374	. 2 |- ( ( ( ch <-> ps ) <-> ( ph <-> ch ) ) <-> ( ch <-> ( ps <-> ( ph <-> ch ) ) ) )
8	6, 7	bitr4i 267	1 |- ( ( ph <-> ps ) <-> ( ( ch <-> ps ) <-> ( ph <-> ch ) ) )

Syntax hints:   <-> wb 196

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

This theorem is referenced by: (None)