Theorem biluk 899

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 191	. . . . 5 |- ((ph <-> ps) <-> (ps <-> ph))
2	1	bibi1i 305	. . . 4 |- (((ph <-> ps) <-> ch) <-> ((ps <-> ph) <-> ch))
3		biass 348	. . . 4 |- (((ps <-> ph) <-> ch) <-> (ps <-> (ph <-> ch)))
4	2, 3	bitri 240	. . 3 |- (((ph <-> ps) <-> ch) <-> (ps <-> (ph <-> ch)))
5		biass 348	. . 3 |- ((((ph <-> ps) <-> ch) <-> (ps <-> (ph <-> ch))) <-> ((ph <-> ps) <-> (ch <-> (ps <-> (ph <-> ch)))))
6	4, 5	mpbi 199	. 2 |- ((ph <-> ps) <-> (ch <-> (ps <-> (ph <-> ch))))
7		biass 348	. 2 |- (((ch <-> ps) <-> (ph <-> ch)) <-> (ch <-> (ps <-> (ph <-> ch))))
8	6, 7	bitr4i 243	1 |- ((ph <-> ps) <-> ((ch <-> ps) <-> (ph <-> ch)))

Syntax hints:   <-> wb 176

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

This theorem is referenced by: (None)