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Metamath Proof Explorer

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-ax1 ⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
2 tbw-ax2 ⊢ ( 𝜓 → ( ( 𝜓 → 𝜒 ) → 𝜓 ) )
3 tbw-ax1 ⊢ ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )
4 2 3 tbwsyl ⊢ ( 𝜓 → ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )
5 tbw-ax1 ⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) → ( ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) )
6 tbw-ax3 ⊢ ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) )
7 5 6 tbwsyl ⊢ ( ( ( 𝜓 → 𝜒 ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) → ( ( 𝜓 → 𝜒 ) → 𝜒 ) )
8 4 7 tbwsyl ⊢ ( 𝜓 → ( ( 𝜓 → 𝜒 ) → 𝜒 ) )
9 tbw-ax1 ⊢ ( ( 𝜓 → ( ( 𝜓 → 𝜒 ) → 𝜒 ) ) → ( ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
10 8 9 ax-mp ⊢ ( ( ( ( 𝜓 → 𝜒 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
11 1 10 tbwsyl ⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )