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Theorem biluk 973
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 212 . . . . 5 ((𝜑𝜓) ↔ (𝜓𝜑))
21bibi1i 328 . . . 4 (((𝜑𝜓) ↔ 𝜒) ↔ ((𝜓𝜑) ↔ 𝜒))
3 biass 374 . . . 4 (((𝜓𝜑) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑𝜒)))
42, 3bitri 264 . . 3 (((𝜑𝜓) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑𝜒)))
5 biass 374 . . 3 ((((𝜑𝜓) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑𝜒))) ↔ ((𝜑𝜓) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑𝜒)))))
64, 5mpbi 220 . 2 ((𝜑𝜓) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑𝜒))))
7 biass 374 . 2 (((𝜒𝜓) ↔ (𝜑𝜒)) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑𝜒))))
86, 7bitr4i 267 1 ((𝜑𝜓) ↔ ((𝜒𝜓) ↔ (𝜑𝜒)))
Colors of variables: wff setvar class
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)
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