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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 → ⊥ ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) )

Proof

Step Hyp Ref Expression
1 tbw-ax1 ⊢ ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ( ( ( 𝜓 → ⊥ ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) )
2 tbw-ax4 ⊢ ( ⊥ → 𝜒 )
3 tbw-ax1 ⊢ ( ( 𝜓 → ⊥ ) → ( ( ⊥ → 𝜒 ) → ( 𝜓 → 𝜒 ) ) )
4 tbwlem1 ⊢ ( ( ( 𝜓 → ⊥ ) → ( ( ⊥ → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ⊥ → 𝜒 ) → ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) ) ) )
5 3 4 ax-mp ⊢ ( ( ⊥ → 𝜒 ) → ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) ) )
6 2 5 ax-mp ⊢ ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) )
7 tbwlem1 ⊢ ( ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( ( 𝜓 → ⊥ ) → 𝜒 ) ) )
8 6 7 ax-mp ⊢ ( 𝜓 → ( ( 𝜓 → ⊥ ) → 𝜒 ) )
9 tbw-ax1 ⊢ ( ( 𝜓 → ( ( 𝜓 → ⊥ ) → 𝜒 ) ) → ( ( ( ( 𝜓 → ⊥ ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
10 8 9 ax-mp ⊢ ( ( ( ( 𝜓 → ⊥ ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
11 1 10 tbwsyl ⊢ ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )
12 tbw-ax1 ⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) )
13 11 12 tbwsyl ⊢ ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) )