Theorem mercolem4 1773

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

Proof of Theorem mercolem4

Step	Hyp	Ref	Expression
1		merco2 1769	. 2 ⊢ (((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑))))
2		merco2 1769	. . . 4 ⊢ ((((𝜂 → 𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃)) → ((𝜃 → (𝜂 → 𝜑)) → (((𝜃 → 𝜒) → 𝜑) → (𝜏 → (𝜂 → 𝜑)))))
3		merco2 1769	. . . . . . . . 9 ⊢ (((𝜑 → 𝜑) → ((⊥ → 𝜑) → (𝜃 → 𝜒))) → (((𝜃 → 𝜒) → 𝜑) → (𝜏 → (𝜂 → 𝜑))))
4		mercolem1 1770	. . . . . . . . 9 ⊢ ((((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 5	. . . . . . . 8 ⊢ (((⊥ → 𝜑) → (𝜃 → 𝜒)) → ((𝜃 → (𝜂 → 𝜑)) → (((𝜃 → 𝜒) → 𝜑) → (𝜏 → (𝜂 → 𝜑)))))
6		mercolem1 1770	. . . . . . . 8 ⊢ ((((⊥ → 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 5	. . . . . . 7 ⊢ ((𝜃 → 𝜒) → ((⊥ → 𝜑) → ((𝜃 → (𝜂 → 𝜑)) → (((𝜃 → 𝜒) → 𝜑) → (𝜏 → (𝜂 → 𝜑))))))
8		merco2 1769	. . . . . . 7 ⊢ (((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 5	. . . . . 6 ⊢ ((((𝜃 → (𝜂 → 𝜑)) → (((𝜃 → 𝜒) → 𝜑) → (𝜏 → (𝜂 → 𝜑)))) → 𝜃) → (((𝜂 → 𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃)))
10		mercolem3 1772	. . . . . 6 ⊢ (((((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 5	. . . . 5 ⊢ ((((𝜃 → (𝜂 → 𝜑)) → (((𝜃 → 𝜒) → 𝜑) → (𝜏 → (𝜂 → 𝜑)))) → 𝜃) → ((⊥ → 𝜑) → (((𝜂 → 𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃))))
12		merco2 1769	. . . . 5 ⊢ (((((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 5	. . . 4 ⊢ (((((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 5	. . 3 ⊢ ((((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 5	. 2 ⊢ ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((𝜃 → (𝜂 → 𝜑)) → (((𝜃 → 𝜒) → 𝜑) → (𝜏 → (𝜂 → 𝜑)))))
16	1, 15	ax-mp 5	1 ⊢ ((𝜃 → (𝜂 → 𝜑)) → (((𝜃 → 𝜒) → 𝜑) → (𝜏 → (𝜂 → 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