Theorem mercolem1 1738

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1737. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mercolem1	⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))

Proof of Theorem mercolem1

Step	Hyp	Ref	Expression
1		merco2 1737	. 2 ⊢ (((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑))))
2		merco2 1737	. . . 4 ⊢ (((𝜒 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒))))
3		merco2 1737	. . . . . . 7 ⊢ (((𝜓 → (𝜃 → 𝜒)) → ((⊥ → 𝜑) → ⊥)) → ((⊥ → 𝜓) → ((⊥ → 𝜑) → (𝜑 → 𝜓))))
4		merco2 1737	. . . . . . 7 ⊢ ((((𝜓 → (𝜃 → 𝜒)) → ((⊥ → 𝜑) → ⊥)) → ((⊥ → 𝜓) → ((⊥ → 𝜑) → (𝜑 → 𝜓)))) → ((((⊥ → 𝜑) → (𝜑 → 𝜓)) → (𝜓 → (𝜃 → 𝜒))) → ((⊥ → 𝜑) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒))))))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ ((((⊥ → 𝜑) → (𝜑 → 𝜓)) → (𝜓 → (𝜃 → 𝜒))) → ((⊥ → 𝜑) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))))
6		merco2 1737	. . . . . 6 ⊢ (((((⊥ → 𝜑) → (𝜑 → 𝜓)) → (𝜓 → (𝜃 → 𝜒))) → ((⊥ → 𝜑) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒))))) → (((((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒))) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → ((⊥ → 𝜑) → ((𝜒 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))))))
7	5, 6	ax-mp 5	. . . . 5 ⊢ (((((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒))) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → ((⊥ → 𝜑) → ((𝜒 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓)))))
8		merco2 1737	. . . . 5 ⊢ ((((((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒))) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → ((⊥ → 𝜑) → ((𝜒 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))))) → ((((𝜒 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))))))
9	7, 8	ax-mp 5	. . . 4 ⊢ ((((𝜒 → 𝜑) → ((⊥ → 𝜑) → (𝜑 → 𝜓))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒))))))
10	2, 9	ax-mp 5	. . 3 ⊢ ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))))
11	1, 10	ax-mp 5	. 2 ⊢ ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒))))
12	1, 11	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → (𝜃 → 𝜒)))

Syntax hints:   → wi 4  ⊥wfal 1549

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 1540  df-fal 1550

This theorem is referenced by:  mercolem4  1741  mercolem5  1742  mercolem6  1743  re1tbw2  1747