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 ) )