Theorem mercolem5 1774

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1769. (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 1769	. 2 ⊢ (((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑))))
2		merco2 1769	. . . . 5 ⊢ (((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜃)) → ((𝜃 → 𝜑) → (𝜏 → (𝜒 → 𝜑))))
3		mercolem1 1770	. . . . 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 1771	. . . . 5 ⊢ (((𝜃 → ((𝜃 → 𝜑) → (𝜏 → (𝜒 → 𝜑)))) → 𝜃) → ((⊥ → 𝜑) → ((⊥ → 𝜑) → 𝜃)))
6		merco2 1769	. . . . 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 1582

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

This proof depends on definitions:  df-bi 210  df-tru 1573  df-fal 1583

This theorem is used by:  mercolem6  1775  mercolem7  1776