Theorem mercolem8 1748

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1740. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mercolem8	⊢ ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒)))))

Proof of Theorem mercolem8

Step	Hyp	Ref	Expression
1		merco2 1740	. 2 ⊢ (((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑))))
2		merco2 1740	. . . . 5 ⊢ ((((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒)))))
3		mercolem3 1743	. . . . 5 ⊢ (((((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒))))) → ((((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒)))))))
4	2, 3	ax-mp 5	. . . 4 ⊢ ((((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒))))))
5		mercolem7 1747	. . . . . 6 ⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))
6		mercolem7 1747	. . . . . 6 ⊢ (((𝜑 → 𝜓) → (((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓))) → ((((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒))))) → ((⊥ → 𝜑) → (((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))) → ((⊥ → 𝜑) → (((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))))
7	5, 6	ax-mp 5	. . . . 5 ⊢ ((((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒))))) → ((⊥ → 𝜑) → (((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))) → ((⊥ → 𝜑) → (((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓))))
8		merco2 1740	. . . . 5 ⊢ (((((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒))))) → ((⊥ → 𝜑) → (((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))) → ((⊥ → 𝜑) → (((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))) → (((((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒)))))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒)))))))))
9	7, 8	ax-mp 5	. . . 4 ⊢ (((((𝜑 → 𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒)))))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒))))))))
10	4, 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) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒)))))))
11	1, 10	ax-mp 5	. 2 ⊢ ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒))))))
12	1, 11	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → ((𝜓 → (𝜑 → 𝜒)) → (𝜏 → (𝜃 → (𝜑 → 𝜒)))))

Syntax hints:   → wi 4  ⊥wfal 1551

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 1542  df-fal 1552

This theorem is referenced by:  re1tbw1  1749