Theorem mercolem4

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	mercolem4	⊢ ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		merco2	⊢ ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) )
2		merco2	⊢ ( ( ( ( 𝜂 → 𝜑 ) → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜃 ) ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) )
3		merco2	⊢ ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → ( 𝜃 → 𝜒 ) ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) )
4		mercolem1	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → ( 𝜃 → 𝜒 ) ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) → ( ( ( ⊥ → 𝜑 ) → ( 𝜃 → 𝜒 ) ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) ) )
5	3 4	ax-mp	⊢ ( ( ( ⊥ → 𝜑 ) → ( 𝜃 → 𝜒 ) ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) )
6		mercolem1	⊢ ( ( ( ( ⊥ → 𝜑 ) → ( 𝜃 → 𝜒 ) ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) ) → ( ( 𝜃 → 𝜒 ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) ) ) )
7	5 6	ax-mp	⊢ ( ( 𝜃 → 𝜒 ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) ) )
8		merco2	⊢ ( ( ( 𝜃 → 𝜒 ) → ( ( ⊥ → 𝜑 ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) ) ) → ( ( ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) → 𝜃 ) → ( ( ( 𝜂 → 𝜑 ) → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜃 ) ) ) )
9	7 8	ax-mp	⊢ ( ( ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) → 𝜃 ) → ( ( ( 𝜂 → 𝜑 ) → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜃 ) ) )
10		mercolem3	⊢ ( ( ( ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) → 𝜃 ) → ( ( ( 𝜂 → 𝜑 ) → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜃 ) ) ) → ( ( ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) → 𝜃 ) → ( ( ⊥ → 𝜑 ) → ( ( ( 𝜂 → 𝜑 ) → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜃 ) ) ) ) )
11	9 10	ax-mp	⊢ ( ( ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) → 𝜃 ) → ( ( ⊥ → 𝜑 ) → ( ( ( 𝜂 → 𝜑 ) → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜃 ) ) ) )
12		merco2	⊢ ( ( ( ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) → 𝜃 ) → ( ( ⊥ → 𝜑 ) → ( ( ( 𝜂 → 𝜑 ) → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜃 ) ) ) ) → ( ( ( ( ( 𝜂 → 𝜑 ) → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜃 ) ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) ) ) ) )
13	11 12	ax-mp	⊢ ( ( ( ( ( 𝜂 → 𝜑 ) → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜃 ) ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) ) ) )
14	2 13	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) ) )
15	1 14	ax-mp	⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) ) )
16	1 15	ax-mp	⊢ ( ( 𝜃 → ( 𝜂 → 𝜑 ) ) → ( ( ( 𝜃 → 𝜒 ) → 𝜑 ) → ( 𝜏 → ( 𝜂 → 𝜑 ) ) ) )