New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  biluk GIF version

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 ((φψ) ↔ ((χψ) ↔ (φχ)))

Proof of Theorem biluk
StepHypRef Expression
1 bicom 191 . . . . 5 ((φψ) ↔ (ψφ))
21bibi1i 305 . . . 4 (((φψ) ↔ χ) ↔ ((ψφ) ↔ χ))
3 biass 348 . . . 4 (((ψφ) ↔ χ) ↔ (ψ ↔ (φχ)))
42, 3bitri 240 . . 3 (((φψ) ↔ χ) ↔ (ψ ↔ (φχ)))
5 biass 348 . . 3 ((((φψ) ↔ χ) ↔ (ψ ↔ (φχ))) ↔ ((φψ) ↔ (χ ↔ (ψ ↔ (φχ)))))
64, 5mpbi 199 . 2 ((φψ) ↔ (χ ↔ (ψ ↔ (φχ))))
7 biass 348 . 2 (((χψ) ↔ (φχ)) ↔ (χ ↔ (ψ ↔ (φχ))))
86, 7bitr4i 243 1 ((φψ) ↔ ((χψ) ↔ (φχ)))
 Colors of variables: wff setvar class 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)
 Copyright terms: Public domain W3C validator