Theorem tbwlem1

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	tbwlem1	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tbw-ax2	⊢ ( 𝜓 → ( ( 𝜓 → 𝜒 ) → 𝜓 ) )
2		tbw-ax1	⊢ ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )
3	1 2	tbwsyl	⊢ ( 𝜓 → ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )
4		tbw-ax1	⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) → ( ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )
5		tbw-ax3	⊢ ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) )
6	4 5	tbwsyl	⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) )
7	3 6	tbwsyl	⊢ ( 𝜓 → ( ( 𝜓 → 𝜒 ) → 𝜒 ) )
8		tbw-ax1	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
9		tbw-ax1	⊢ ( ( 𝜓 → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) → ( ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
10	7 8 9	mpsyl	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )