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Metamath Proof Explorer

Theorem tbwlem4

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 tbwlem4 ( ( ( 𝜑 → ⊥ ) → 𝜓 ) → ( ( 𝜓 → ⊥ ) → 𝜑 ) )

Proof

Step Hyp Ref Expression
1 tbw-ax4 ⊢ ( ⊥ → ⊥ )
2 tbw-ax1 ⊢ ( ( 𝜓 → ⊥ ) → ( ( ⊥ → ⊥ ) → ( 𝜓 → ⊥ ) ) )
3 tbwlem1 ⊢ ( ( ( 𝜓 → ⊥ ) → ( ( ⊥ → ⊥ ) → ( 𝜓 → ⊥ ) ) ) → ( ( ⊥ → ⊥ ) → ( ( 𝜓 → ⊥ ) → ( 𝜓 → ⊥ ) ) ) )
4 2 3 ax-mp ⊢ ( ( ⊥ → ⊥ ) → ( ( 𝜓 → ⊥ ) → ( 𝜓 → ⊥ ) ) )
5 1 4 ax-mp ⊢ ( ( 𝜓 → ⊥ ) → ( 𝜓 → ⊥ ) )
6 tbwlem1 ⊢ ( ( ( 𝜓 → ⊥ ) → ( 𝜓 → ⊥ ) ) → ( 𝜓 → ( ( 𝜓 → ⊥ ) → ⊥ ) ) )
7 5 6 ax-mp ⊢ ( 𝜓 → ( ( 𝜓 → ⊥ ) → ⊥ ) )
8 tbw-ax1 ⊢ ( ( ( 𝜑 → ⊥ ) → 𝜓 ) → ( ( 𝜓 → ( ( 𝜓 → ⊥ ) → ⊥ ) ) → ( ( 𝜑 → ⊥ ) → ( ( 𝜓 → ⊥ ) → ⊥ ) ) ) )
9 tbwlem1 ⊢ ( ( ( ( 𝜑 → ⊥ ) → 𝜓 ) → ( ( 𝜓 → ( ( 𝜓 → ⊥ ) → ⊥ ) ) → ( ( 𝜑 → ⊥ ) → ( ( 𝜓 → ⊥ ) → ⊥ ) ) ) ) → ( ( 𝜓 → ( ( 𝜓 → ⊥ ) → ⊥ ) ) → ( ( ( 𝜑 → ⊥ ) → 𝜓 ) → ( ( 𝜑 → ⊥ ) → ( ( 𝜓 → ⊥ ) → ⊥ ) ) ) ) )
10 8 9 ax-mp ⊢ ( ( 𝜓 → ( ( 𝜓 → ⊥ ) → ⊥ ) ) → ( ( ( 𝜑 → ⊥ ) → 𝜓 ) → ( ( 𝜑 → ⊥ ) → ( ( 𝜓 → ⊥ ) → ⊥ ) ) ) )
11 7 10 ax-mp ⊢ ( ( ( 𝜑 → ⊥ ) → 𝜓 ) → ( ( 𝜑 → ⊥ ) → ( ( 𝜓 → ⊥ ) → ⊥ ) ) )
12 tbwlem2 ⊢ ( ( ( 𝜑 → ⊥ ) → ( ( 𝜓 → ⊥ ) → ⊥ ) ) → ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → ( ( 𝜓 → ⊥ ) → 𝜑 ) ) )
13 tbwlem3 ⊢ ( ( ( ( ( 𝜑 → ⊥ ) → 𝜑 ) → 𝜑 ) → ( ( 𝜓 → ⊥ ) → 𝜑 ) ) → ( ( 𝜓 → ⊥ ) → 𝜑 ) )
14 12 13 tbwsyl ⊢ ( ( ( 𝜑 → ⊥ ) → ( ( 𝜓 → ⊥ ) → ⊥ ) ) → ( ( 𝜓 → ⊥ ) → 𝜑 ) )
15 11 14 tbwsyl ⊢ ( ( ( 𝜑 → ⊥ ) → 𝜓 ) → ( ( 𝜓 → ⊥ ) → 𝜑 ) )