Theorem biluk 743

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

Assertion

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

Proof of Theorem biluk

Step	Hyp	Ref	Expression
1		bicom 518	. . . . 5 |- ((ph <-> ps) <-> (ps <-> ph))
2	1	bibi1i 607	. . . 4 |- (((ph <-> ps) <-> ch) <-> ((ps <-> ph) <-> ch))
3		biass 742	. . . 4 |- (((ps <-> ph) <-> ch) <-> (ps <-> (ph <-> ch)))
4	2, 3	bitr 173	. . 3 |- (((ph <-> ps) <-> ch) <-> (ps <-> (ph <-> ch)))
5		biass 742	. . 3 |- ((((ph <-> ps) <-> ch) <-> (ps <-> (ph <-> ch))) <-> ((ph <-> ps) <-> (ch <-> (ps <-> (ph <-> ch)))))
6	4, 5	mpbi 189	. 2 |- ((ph <-> ps) <-> (ch <-> (ps <-> (ph <-> ch))))
7		biass 742	. 2 |- (((ch <-> ps) <-> (ph <-> ch)) <-> (ch <-> (ps <-> (ph <-> ch))))
8	6, 7	bitr4 176	1 |- ((ph <-> ps) <-> ((ch <-> ps) <-> (ph <-> ch)))

Syntax hints:   <-> wb 146

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-or 224  df-an 225

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