Theorem merco1lem18

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 . (Contributed by Anthony Hart, 18-Sep-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	merco1lem18	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		merco1	⊢ ( ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ( ( 𝜓 → 𝜒 ) → 𝜓 ) ) → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) )
2		merco1lem17	⊢ ( ( ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ( ( 𝜓 → 𝜒 ) → 𝜓 ) ) → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) )
3	1 2	ax-mp	⊢ ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) )
4		merco1lem17	⊢ ( ( ( ( ( 𝜓 → 𝜒 ) → 𝜓 ) → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) )
5	3 4	ax-mp	⊢ ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) )
6		merco1lem5	⊢ ( ( ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) → ⊥ ) → ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) )
7		merco1lem3	⊢ ( ( ( ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) → ⊥ ) → ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) ) → ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) )
8	6 7	ax-mp	⊢ ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) )
9		merco1lem5	⊢ ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) → ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) )
10	8 9	ax-mp	⊢ ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) )
11		merco1lem4	⊢ ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) → ( ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) )
12	10 11	ax-mp	⊢ ( ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) )
13		merco1	⊢ ( ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ( 𝜓 → 𝜑 ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) ) )
14		merco1lem2	⊢ ( ( ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ( 𝜓 → 𝜑 ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) ) ) → ( ( ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) ) ) )
15	13 14	ax-mp	⊢ ( ( ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) → ( ( ( ( ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) → ⊥ ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ⊥ ) ) → ⊥ ) → ⊥ ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) ) )
16	12 15	ax-mp	⊢ ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) )
17		merco1lem9	⊢ ( ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) ) → ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) )
18	16 17	ax-mp	⊢ ( ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) ) )
19	5 18	ax-mp	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜓 → 𝜑 ) → ( 𝜓 → 𝜒 ) ) )