Theorem mercolem2 1738

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1736. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mercolem2	⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))

Proof of Theorem mercolem2

Step	Hyp	Ref	Expression
1		merco2 1736	. 2 ⊢ (((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑))))
2		merco2 1736	. . . 4 ⊢ (((𝜑 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑))))
3		merco2 1736	. . . . . . . 8 ⊢ (((𝜑 → 𝜓) → ((⊥ → 𝜑) → ⊥)) → ((⊥ → 𝜑) → (𝜒 → (𝜃 → 𝜑))))
4		merco2 1736	. . . . . . . 8 ⊢ ((((𝜑 → 𝜓) → ((⊥ → 𝜑) → ⊥)) → ((⊥ → 𝜑) → (𝜒 → (𝜃 → 𝜑)))) → (((𝜒 → (𝜃 → 𝜑)) → (𝜑 → 𝜓)) → ((⊥ → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓)))))
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ (((𝜒 → (𝜃 → 𝜑)) → (𝜑 → 𝜓)) → ((⊥ → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))))
6		merco2 1736	. . . . . . 7 ⊢ ((((𝜒 → (𝜃 → 𝜑)) → (𝜑 → 𝜓)) → ((⊥ → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓)))) → ((((⊥ → 𝜑) → (𝜑 → 𝜓)) → (𝜒 → (𝜃 → 𝜑))) → ((⊥ → 𝜑) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑))))))
7	5, 6	ax-mp 5	. . . . . 6 ⊢ ((((⊥ → 𝜑) → (𝜑 → 𝜓)) → (𝜒 → (𝜃 → 𝜑))) → ((⊥ → 𝜑) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))))
8		merco2 1736	. . . . . 6 ⊢ (((((⊥ → 𝜑) → (𝜑 → 𝜓)) → (𝜒 → (𝜃 → 𝜑))) → ((⊥ → 𝜑) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑))))) → (((((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑))) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → ((⊥ → 𝜑) → ((𝜑 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))))))
9	7, 8	ax-mp 5	. . . . 5 ⊢ (((((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑))) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → ((⊥ → 𝜑) → ((𝜑 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓)))))
10		merco2 1736	. . . . 5 ⊢ ((((((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑))) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → ((⊥ → 𝜑) → ((𝜑 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))))))
11	9, 10	ax-mp 5	. . . 4 ⊢ ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑))))))
12	2, 11	ax-mp 5	. . 3 ⊢ ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))))
13	1, 12	ax-mp 5	. 2 ⊢ ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑))))
14	1, 13	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → 𝜑) → (𝜒 → (𝜃 → 𝜑)))

Syntax hints:   → wi 4  ⊥wfal 1548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 209  df-tru 1539  df-fal 1549

This theorem is referenced by:  mercolem3  1739  mercolem5  1741  re1tbw3  1747