Metamath Proof Explorer


Theorem tbwlem5

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 tbwlem5 ( ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ⊥ ) → 𝜑 )

Proof

Step Hyp Ref Expression
1 tbw-ax2 ( 𝜑 → ( 𝜓𝜑 ) )
2 tbw-ax1 ( ( 𝜓𝜑 ) → ( ( 𝜑 → ⊥ ) → ( 𝜓 → ⊥ ) ) )
3 1 2 tbwsyl ( 𝜑 → ( ( 𝜑 → ⊥ ) → ( 𝜓 → ⊥ ) ) )
4 tbwlem1 ( ( 𝜑 → ( ( 𝜑 → ⊥ ) → ( 𝜓 → ⊥ ) ) ) → ( ( 𝜑 → ⊥ ) → ( 𝜑 → ( 𝜓 → ⊥ ) ) ) )
5 3 4 ax-mp ( ( 𝜑 → ⊥ ) → ( 𝜑 → ( 𝜓 → ⊥ ) ) )
6 tbwlem4 ( ( ( 𝜑 → ⊥ ) → ( 𝜑 → ( 𝜓 → ⊥ ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ⊥ ) → 𝜑 ) )
7 5 6 ax-mp ( ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ⊥ ) → 𝜑 )