Theorem tbwlem2

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	tbwlem2	⊢ ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tbw-ax1	⊢ ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ( ( ( 𝜓 → ⊥ ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
2		tbw-ax4	⊢ ( ⊥ → 𝜒 )
3		tbw-ax1	⊢ ( ( 𝜓 → ⊥ ) → ( ( ⊥ → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
4		tbwlem1	⊢ ( ( ( 𝜓 → ⊥ ) → ( ( ⊥ → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ⊥ → 𝜒 ) → ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) ) ) )
5	3 4	ax-mp	⊢ ( ( ⊥ → 𝜒 ) → ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) ) )
6	2 5	ax-mp	⊢ ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) )
7		tbwlem1	⊢ ( ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( ( 𝜓 → ⊥ ) → 𝜒 ) ) )
8	6 7	ax-mp	⊢ ( 𝜓 → ( ( 𝜓 → ⊥ ) → 𝜒 ) )
9		tbw-ax1	⊢ ( ( 𝜓 → ( ( 𝜓 → ⊥ ) → 𝜒 ) ) → ( ( ( ( 𝜓 → ⊥ ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
10	8 9	ax-mp	⊢ ( ( ( ( 𝜓 → ⊥ ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
11	1 10	tbwsyl	⊢ ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
12		tbw-ax1	⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) )
13	11 12	tbwsyl	⊢ ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) )