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Metamath Proof Explorer

Theorem re1luk2

Description: luk-2 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	re1luk2	⊢ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		tbw-negdf	⊢ ( ( ( ¬ 𝜑 → ( 𝜑 → ⊥ ) ) → ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ⊥ ) ) → ⊥ )
2		tbw-ax2	⊢ ( ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ⊥ ) → ( ( ¬ 𝜑 → ( 𝜑 → ⊥ ) ) → ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ⊥ ) ) )
3		tbwlem4	⊢ ( ( ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ⊥ ) → ( ( ¬ 𝜑 → ( 𝜑 → ⊥ ) ) → ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ⊥ ) ) ) → ( ( ( ( ¬ 𝜑 → ( 𝜑 → ⊥ ) ) → ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ⊥ ) ) → ⊥ ) → ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( ¬ 𝜑 → ( 𝜑 → ⊥ ) ) → ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ⊥ ) ) → ⊥ ) → ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) )
5	1 4	ax-mp	⊢ ( ( 𝜑 → ⊥ ) → ¬ 𝜑 )
6		tbw-ax1	⊢ ( ( ( 𝜑 → ⊥ ) → ¬ 𝜑 ) → ( ( ¬ 𝜑 → 𝜑 ) → ( ( 𝜑 → ⊥ ) → 𝜑 ) ) )
7	5 6	ax-mp	⊢ ( ( ¬ 𝜑 → 𝜑 ) → ( ( 𝜑 → ⊥ ) → 𝜑 ) )
8		tbw-ax3	⊢ ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 )
9	7 8	tbwsyl	⊢ ( ( ¬ 𝜑 → 𝜑 ) → 𝜑 )