Theorem mercolem7 1747

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
mercolem7	⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓)))

Proof of Theorem mercolem7

Step	Hyp	Ref	Expression
1		merco2 1740	. 2 ⊢ (((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑))))
2		mercolem3 1743	. . . 4 ⊢ (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → ((𝜑 → 𝜒) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓))))
3		mercolem6 1746	. . . 4 ⊢ ((((𝜑 → 𝜒) → (𝜃 → 𝜓)) → ((𝜑 → 𝜒) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓)))) → ((𝜑 → 𝜒) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓))))
4	2, 3	ax-mp 5	. . 3 ⊢ ((𝜑 → 𝜒) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓)))
5		mercolem5 1745	. . . 4 ⊢ (𝜑 → ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓))))
6		mercolem4 1744	. . . 4 ⊢ ((𝜑 → ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓)))) → (((𝜑 → 𝜒) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓))))))
7	5, 6	ax-mp 5	. . 3 ⊢ (((𝜑 → 𝜒) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓))) → ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓)))))
8	4, 7	ax-mp 5	. 2 ⊢ ((((𝜑 → 𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑 → 𝜑) → (𝜑 → (𝜑 → 𝜑)))) → ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓))))
9	1, 8	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → (((𝜑 → 𝜒) → (𝜃 → 𝜓)) → (𝜃 → 𝜓)))

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:  mercolem8  1748