Theorem mercolem6

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 . (Contributed by Anthony Hart, 16-Aug-2011) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	mercolem6	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		merco2	⊢ ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) )
2		mercolem1	⊢ ( ( ( 𝜑 → ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
3		mercolem1	⊢ ( ( ( ( 𝜑 → ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
4	2 3	ax-mp	⊢ ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) )
5		mercolem5	⊢ ( 𝜑 → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
6		mercolem4	⊢ ( ( 𝜑 → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) )
7	5 6	ax-mp	⊢ ( ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) )
8	4 7	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
9	1 8	ax-mp	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) )
10		mercolem1	⊢ ( ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
11		mercolem1	⊢ ( ( ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) )
12	10 11	ax-mp	⊢ ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) )
13		mercolem5	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) )
14		mercolem4	⊢ ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) → ( ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) ) )
15	13 14	ax-mp	⊢ ( ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) )
16	12 15	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) )
17	1 16	ax-mp	⊢ ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) )
18	9 17	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) )
19	1 18	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) )
20	1 19	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) )
21	1 20	ax-mp	⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) )