Theorem merco1lem17

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	merco1lem17	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜏 ) → ( ( 𝜑 → 𝜒 ) → 𝜏 ) )

Proof

Step	Hyp	Ref	Expression
1		merco1lem11	⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) )
2		merco1lem7	⊢ ( ( ( ( ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) → 𝜑 ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) )
3		merco1	⊢ ( ( ( ( ( ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) → 𝜑 ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) ) )
4	2 3	ax-mp	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) )
5		merco1lem9	⊢ ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) )
6	4 5	ax-mp	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) ) → ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) )
7	1 6	ax-mp	⊢ ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 )
8		merco1	⊢ ( ( ( ( ( 𝜒 → 𝜑 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → ⊥ ) ) → ⊥ ) → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) )
9	7 8	ax-mp	⊢ ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) )
10		merco1lem11	⊢ ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) )
11		merco1lem7	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) )
12		merco1	⊢ ( ( ( ( ( ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → 𝜑 ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ⊥ ) ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) ) )
13	11 12	ax-mp	⊢ ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) )
14		merco1lem9	⊢ ( ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) )
15	13 14	ax-mp	⊢ ( ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) )
16	10 15	ax-mp	⊢ ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) )
17		merco1	⊢ ( ( ( ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜑 ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ⊥ ) ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) ) )
18	16 17	ax-mp	⊢ ( ( ( 𝜑 → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) )
19	9 18	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) )
20		merco1lem4	⊢ ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) )
21		merco1lem16	⊢ ( ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) → ( ( ( ( 𝜑 → 𝜒 ) → ⊥ ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) )
22	19 20 21	mpsyl	⊢ ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) )
23		merco1	⊢ ( ( ( ( ( 𝜏 → 𝜑 ) → ( ( 𝜑 → 𝜒 ) → ⊥ ) ) → 𝜒 ) → ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) ) → ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜏 ) → ( ( 𝜑 → 𝜒 ) → 𝜏 ) ) )
24	22 23	ax-mp	⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜒 ) → 𝜏 ) → ( ( 𝜑 → 𝜒 ) → 𝜏 ) )