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Metamath Proof Explorer

Theorem tbwlem3

Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	tbwlem3	⊢ ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 )

Proof

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