Theorem biluk 512

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 398	. . . . 5 |- ((ph <-> ch) <-> (ch <-> ph))
2		biass 511	. . . . 5 |- (((ph <-> ch) <-> (ch <-> ph)) <-> (ph <-> (ch <-> (ch <-> ph))))
3	1, 2	mpbi 164	. . . 4 |- (ph <-> (ch <-> (ch <-> ph)))
4	3	bibi2i 460	. . 3 |- ((ps <-> ph) <-> (ps <-> (ch <-> (ch <-> ph))))
5		bicom 398	. . 3 |- ((ph <-> ps) <-> (ps <-> ph))
6		biass 511	. . 3 |- (((ps <-> ch) <-> (ch <-> ph)) <-> (ps <-> (ch <-> (ch <-> ph))))
7	4, 5, 6	3bitr4 158	. 2 |- ((ph <-> ps) <-> ((ps <-> ch) <-> (ch <-> ph)))
8		bicom 398	. . 3 |- ((ps <-> ch) <-> (ch <-> ps))
9		bicom 398	. . 3 |- ((ch <-> ph) <-> (ph <-> ch))
10	8, 9	bibi12i 462	. 2 |- (((ps <-> ch) <-> (ch <-> ph)) <-> ((ch <-> ps) <-> (ph <-> ch)))
11	7, 10	bitr 151	1 |- ((ph <-> ps) <-> ((ch <-> ps) <-> (ph <-> ch)))

Syntax hints:   <-> wb 127

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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