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Metamath Proof Explorer

Theorem biluk

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)

		Ref	Expression
	Assertion	biluk	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜒 ↔ 𝜓 ) ↔ ( 𝜑 ↔ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bicom	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜓 ↔ 𝜑 ) )
2	1	bibi1i	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ↔ ( ( 𝜓 ↔ 𝜑 ) ↔ 𝜒 ) )
3		biass	⊢ ( ( ( 𝜓 ↔ 𝜑 ) ↔ 𝜒 ) ↔ ( 𝜓 ↔ ( 𝜑 ↔ 𝜒 ) ) )
4	2 3	bitri	⊢ ( ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ↔ ( 𝜓 ↔ ( 𝜑 ↔ 𝜒 ) ) )
5		biass	⊢ ( ( ( ( 𝜑 ↔ 𝜓 ) ↔ 𝜒 ) ↔ ( 𝜓 ↔ ( 𝜑 ↔ 𝜒 ) ) ) ↔ ( ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜒 ↔ ( 𝜓 ↔ ( 𝜑 ↔ 𝜒 ) ) ) ) )
6	4 5	mpbi	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( 𝜒 ↔ ( 𝜓 ↔ ( 𝜑 ↔ 𝜒 ) ) ) )
7		biass	⊢ ( ( ( 𝜒 ↔ 𝜓 ) ↔ ( 𝜑 ↔ 𝜒 ) ) ↔ ( 𝜒 ↔ ( 𝜓 ↔ ( 𝜑 ↔ 𝜒 ) ) ) )
8	6 7	bitr4i	⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( 𝜒 ↔ 𝜓 ) ↔ ( 𝜑 ↔ 𝜒 ) ) )