Theorem merco1lem1

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 17-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	merco1lem1	⊢ ( 𝜑 → ( ⊥ → 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		merco1	⊢ ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( ( ( ⊥ → 𝜑 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) )
2		merco1	⊢ ( ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( ( ( ⊥ → 𝜑 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) ) → ( ( ( ( ( ⊥ → 𝜑 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( ( ( ⊥ → 𝜑 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) )
4		merco1	⊢ ( ( ( ( ( ( ⊥ → 𝜑 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) )
5	3 4	ax-mp	⊢ ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) )
6		merco1	⊢ ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) → ( 𝜑 → ⊥ ) ) )
7		merco1	⊢ ( ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) → ( 𝜑 → ⊥ ) ) ) → ( ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) ) )
8	6 7	ax-mp	⊢ ( ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) )
9		merco1	⊢ ( ( ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) ) → ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) ) )
10	8 9	ax-mp	⊢ ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) )
11	5 10	ax-mp	⊢ ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) )
12		merco1	⊢ ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜒 ) ) → ( ( ( ⊥ → 𝜒 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) )
13		merco1	⊢ ( ( ( ( ( ( ⊥ → 𝜑 ) → ( 𝜑 → ⊥ ) ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜒 ) ) → ( ( ( ⊥ → 𝜒 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) ) → ( ( ( ( ( ⊥ → 𝜒 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) )
14	12 13	ax-mp	⊢ ( ( ( ( ( ⊥ → 𝜒 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) )
15		merco1	⊢ ( ( ( ( ( ( ⊥ → 𝜒 ) → ⊥ ) → ( 𝜑 → ⊥ ) ) → ( ⊥ → 𝜑 ) ) → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) )
16	14 15	ax-mp	⊢ ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) )
17		merco1	⊢ ( ( ( ( ( ⊥ → 𝜒 ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜒 ) ) → ⊥ ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) )
18		merco1	⊢ ( ( ( ( ( ( ⊥ → 𝜒 ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ⊥ ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) → ( ( ( 𝜑 → ( ⊥ → 𝜒 ) ) → ⊥ ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) ) → ( ( ( ( ( 𝜑 → ( ⊥ → 𝜒 ) ) → ⊥ ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) → ( ⊥ → 𝜒 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) ) )
19	17 18	ax-mp	⊢ ( ( ( ( ( 𝜑 → ( ⊥ → 𝜒 ) ) → ⊥ ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) → ( ⊥ → 𝜒 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) )
20		merco1	⊢ ( ( ( ( ( ( 𝜑 → ( ⊥ → 𝜒 ) ) → ⊥ ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ⊥ ) ) → ( ⊥ → 𝜒 ) ) → ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) ) → ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) ) )
21	19 20	ax-mp	⊢ ( ( ( ( 𝜑 → ( ⊥ → 𝜑 ) ) → ( ⊥ → 𝜒 ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) → ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) ) )
22	16 21	ax-mp	⊢ ( ( 𝜑 → ( 𝜑 → ( ⊥ → 𝜑 ) ) ) → ( 𝜑 → ( ⊥ → 𝜒 ) ) )
23	11 22	ax-mp	⊢ ( 𝜑 → ( ⊥ → 𝜒 ) )