Metamath Proof Explorer


Theorem tbwlem3

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 tbwlem3 ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 )

Proof

Step Hyp Ref Expression
1 tbw-ax3 ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 )
2 tbw-ax2 ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) ) )
3 tbw-ax1 ( ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) ) )
4 2 3 tbwsyl ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) ) )
5 1 4 ax-mp ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) )
6 tbw-ax1 ( ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) ) → ( ( ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) → 𝜓 ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) ) )
7 tbw-ax3 ( ( ( ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) → 𝜓 ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) )
8 6 7 tbwsyl ( ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) ) → ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 ) )
9 5 8 ax-mp ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → 𝜓 ) → 𝜓 )