Theorem mercolem5 1506

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mercolem5	⊢ (θ → ((θ → φ) → (τ → (χ → φ))))

Proof of Theorem mercolem5

Step	Hyp	Ref	Expression
1		merco2 1501	. 2 ⊢ (((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ))))
2		merco2 1501	. . . . 5 ⊢ (((φ → φ) → (( ⊥ → φ) → θ)) → ((θ → φ) → (τ → (χ → φ))))
3		mercolem1 1502	. . . . 5 ⊢ ((((φ → φ) → (( ⊥ → φ) → θ)) → ((θ → φ) → (τ → (χ → φ)))) → ((( ⊥ → φ) → θ) → (θ → ((θ → φ) → (τ → (χ → φ))))))
4	2, 3	ax-mp 5	. . . 4 ⊢ ((( ⊥ → φ) → θ) → (θ → ((θ → φ) → (τ → (χ → φ)))))
5		mercolem2 1503	. . . . 5 ⊢ (((θ → ((θ → φ) → (τ → (χ → φ)))) → θ) → (( ⊥ → φ) → (( ⊥ → φ) → θ)))
6		merco2 1501	. . . . 5 ⊢ ((((θ → ((θ → φ) → (τ → (χ → φ)))) → θ) → (( ⊥ → φ) → (( ⊥ → φ) → θ))) → (((( ⊥ → φ) → θ) → (θ → ((θ → φ) → (τ → (χ → φ))))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → (θ → ((θ → φ) → (τ → (χ → φ))))))))
7	5, 6	ax-mp 5	. . . 4 ⊢ (((( ⊥ → φ) → θ) → (θ → ((θ → φ) → (τ → (χ → φ))))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → (θ → ((θ → φ) → (τ → (χ → φ)))))))
8	4, 7	ax-mp 5	. . 3 ⊢ ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → (θ → ((θ → φ) → (τ → (χ → φ))))))
9	1, 8	ax-mp 5	. 2 ⊢ ((((φ → φ) → (( ⊥ → φ) → φ)) → ((φ → φ) → (φ → (φ → φ)))) → (θ → ((θ → φ) → (τ → (χ → φ)))))
10	1, 9	ax-mp 5	1 ⊢ (θ → ((θ → φ) → (τ → (χ → φ))))

This proof depends on syntax axioms:   → wi 4   ⊥ wfal 1317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320

This theorem is used by:  mercolem6  1507  mercolem7  1508