Theorem mercolem3 1745

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1742. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mercolem3	⊢ ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))

Proof of Theorem mercolem3

Step	Hyp	Ref	Expression
1		merco2 1742	. 2 ⊢ (((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑))))
2		merco2 1742	. . . 4 ⊢ (((𝜒 → 𝜑) → ((⊥ → 𝜑) → 𝜓)) → ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒))))
3		mercolem2 1744	. . . . . . 7 ⊢ (((𝜓 → (𝜑 → 𝜒)) → 𝜓) → ((⊥ → 𝜑) → ((⊥ → 𝜑) → 𝜓)))
4		merco2 1742	. . . . . . 7 ⊢ ((((𝜓 → (𝜑 → 𝜒)) → 𝜓) → ((⊥ → 𝜑) → ((⊥ → 𝜑) → 𝜓))) → ((((⊥ → 𝜑) → 𝜓) → (𝜓 → (𝜑 → 𝜒))) → ((⊥ → 𝜑) → ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒))))))
5	3, 4	ax-mp 5	. . . . . 6 ⊢ ((((⊥ → 𝜑) → 𝜓) → (𝜓 → (𝜑 → 𝜒))) → ((⊥ → 𝜑) → ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))))
6		merco2 1742	. . . . . 6 ⊢ (((((⊥ → 𝜑) → 𝜓) → (𝜓 → (𝜑 → 𝜒))) → ((⊥ → 𝜑) → ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒))))) → ((((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒))) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → ((𝜒 → 𝜑) → ((⊥ → 𝜑) → 𝜓)))))
7	5, 6	ax-mp 5	. . . . 5 ⊢ ((((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒))) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → ((𝜒 → 𝜑) → ((⊥ → 𝜑) → 𝜓))))
8		merco2 1742	. . . . 5 ⊢ (((((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒))) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → ((𝜒 → 𝜑) → ((⊥ → 𝜑) → 𝜓)))) → ((((𝜒 → 𝜑) → ((⊥ → 𝜑) → 𝜓)) → ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))))))
9	7, 8	ax-mp 5	. . . 4 ⊢ ((((𝜒 → 𝜑) → ((⊥ → 𝜑) → 𝜓)) → ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 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)))) → ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))))
11	1, 10	ax-mp 5	. 2 ⊢ ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒))))
12	1, 11	ax-mp 5	1 ⊢ ((𝜓 → 𝜒) → (𝜓 → (𝜑 → 𝜒)))

Syntax hints:   → wi 4  ⊥wfal 1553

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 206  df-tru 1544  df-fal 1554

This theorem is referenced by:  mercolem4  1746  mercolem7  1749  mercolem8  1750  re1tbw1  1751  re1tbw4  1754