Metamath Proof Explorer

Theorem mercolem5

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	mercolem5	$⊢ θ \to ((θ \to φ) \to (τ \to (χ \to φ)))$

Proof

Step	Hyp	Ref	Expression
1		merco2	$⊢ ((φ \to φ) \to ((⊥ \to φ) \to φ)) \to ((φ \to φ) \to (φ \to (φ \to φ)))$
2		merco2	$⊢ ((φ \to φ) \to ((⊥ \to φ) \to θ)) \to ((θ \to φ) \to (τ \to (χ \to φ)))$
3		mercolem1	$⊢ (((φ \to φ) \to ((⊥ \to φ) \to θ)) \to ((θ \to φ) \to (τ \to (χ \to φ)))) \to (((⊥ \to φ) \to θ) \to (θ \to ((θ \to φ) \to (τ \to (χ \to φ)))))$
4	2 3	ax-mp	$⊢ ((⊥ \to φ) \to θ) \to (θ \to ((θ \to φ) \to (τ \to (χ \to φ))))$
5		mercolem2	$⊢ ((θ \to ((θ \to φ) \to (τ \to (χ \to φ)))) \to θ) \to ((⊥ \to φ) \to ((⊥ \to φ) \to θ))$
6		merco2	$⊢ (((θ \to ((θ \to φ) \to (τ \to (χ \to φ)))) \to θ) \to ((⊥ \to φ) \to ((⊥ \to φ) \to θ))) \to ((((⊥ \to φ) \to θ) \to (θ \to ((θ \to φ) \to (τ \to (χ \to φ))))) \to ((((φ \to φ) \to ((⊥ \to φ) \to φ)) \to ((φ \to φ) \to (φ \to (φ \to φ)))) \to ((((φ \to φ) \to ((⊥ \to φ) \to φ)) \to ((φ \to φ) \to (φ \to (φ \to φ)))) \to (θ \to ((θ \to φ) \to (τ \to (χ \to φ)))))))$
7	5 6	ax-mp	$⊢ (((⊥ \to φ) \to θ) \to (θ \to ((θ \to φ) \to (τ \to (χ \to φ))))) \to ((((φ \to φ) \to ((⊥ \to φ) \to φ)) \to ((φ \to φ) \to (φ \to (φ \to φ)))) \to ((((φ \to φ) \to ((⊥ \to φ) \to φ)) \to ((φ \to φ) \to (φ \to (φ \to φ)))) \to (θ \to ((θ \to φ) \to (τ \to (χ \to φ))))))$
8	4 7	ax-mp	$⊢ (((φ \to φ) \to ((⊥ \to φ) \to φ)) \to ((φ \to φ) \to (φ \to (φ \to φ)))) \to ((((φ \to φ) \to ((⊥ \to φ) \to φ)) \to ((φ \to φ) \to (φ \to (φ \to φ)))) \to (θ \to ((θ \to φ) \to (τ \to (χ \to φ)))))$
9	1 8	ax-mp	$⊢ (((φ \to φ) \to ((⊥ \to φ) \to φ)) \to ((φ \to φ) \to (φ \to (φ \to φ)))) \to (θ \to ((θ \to φ) \to (τ \to (χ \to φ))))$
10	1 9	ax-mp	$⊢ θ \to ((θ \to φ) \to (τ \to (χ \to φ)))$