Theorem meran2

Description: A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011)

	Ref	Expression
Assertion	meran2	⊢ ( ¬ ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) ∨ ( ¬ ( ¬ 𝜏 ∨ 𝜃 ) ∨ ( 𝜒 ∨ ( 𝜑 ∨ 𝜃 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		meran1	⊢ ( ¬ ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) ∨ ( ¬ ( ¬ 𝜃 ∨ 𝜑 ) ∨ ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) )
2	1	imorri	⊢ ( ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) → ( ¬ ( ¬ 𝜃 ∨ 𝜑 ) ∨ ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) )
3		meran1	⊢ ( ¬ ( ¬ ( ¬ 𝜃 ∨ 𝜑 ) ∨ ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ∨ ( ¬ ( ¬ 𝜏 ∨ 𝜃 ) ∨ ( 𝜒 ∨ ( 𝜑 ∨ 𝜃 ) ) ) )
4	3	imorri	⊢ ( ( ¬ ( ¬ 𝜃 ∨ 𝜑 ) ∨ ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) → ( ¬ ( ¬ 𝜏 ∨ 𝜃 ) ∨ ( 𝜒 ∨ ( 𝜑 ∨ 𝜃 ) ) ) )
5	2 4	syl	⊢ ( ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) → ( ¬ ( ¬ 𝜏 ∨ 𝜃 ) ∨ ( 𝜒 ∨ ( 𝜑 ∨ 𝜃 ) ) ) )
6	5	imori	⊢ ( ¬ ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) ∨ ( ¬ ( ¬ 𝜏 ∨ 𝜃 ) ∨ ( 𝜒 ∨ ( 𝜑 ∨ 𝜃 ) ) ) )