Metamath Proof Explorer


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 → ⊥ ) ) → ( ( ( 𝜑𝜒 ) → 𝜃 ) → ( 𝜓𝜃 ) ) )

Proof

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