Theorem biluk 389

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	⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜒 ↔ 𝜓) ↔ (𝜑 ↔ 𝜒)))

Proof of Theorem biluk

Step	Hyp	Ref	Expression
1		bicom 224	. . . . 5 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
2	1	bibi1i 341	. . . 4 ⊢ (((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ ((𝜓 ↔ 𝜑) ↔ 𝜒))
3		biass 388	. . . 4 ⊢ (((𝜓 ↔ 𝜑) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑 ↔ 𝜒)))
4	2, 3	bitri 277	. . 3 ⊢ (((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑 ↔ 𝜒)))
5		biass 388	. . 3 ⊢ ((((𝜑 ↔ 𝜓) ↔ 𝜒) ↔ (𝜓 ↔ (𝜑 ↔ 𝜒))) ↔ ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑 ↔ 𝜒)))))
6	4, 5	mpbi 232	. 2 ⊢ ((𝜑 ↔ 𝜓) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑 ↔ 𝜒))))
7		biass 388	. 2 ⊢ (((𝜒 ↔ 𝜓) ↔ (𝜑 ↔ 𝜒)) ↔ (𝜒 ↔ (𝜓 ↔ (𝜑 ↔ 𝜒))))
8	6, 7	bitr4i 280	1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜒 ↔ 𝜓) ↔ (𝜑 ↔ 𝜒)))

Syntax hints:   ↔ wb 208

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 209

This theorem is referenced by:  biadan  817