Theorem merco1lem14 1730

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1713. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem14	⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) → (𝜑 → 𝜒))

Proof of Theorem merco1lem14

Step	Hyp	Ref	Expression
1		merco1lem13 1729	. . . 4 ⊢ ((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓)))
2		merco1lem8 1724	. . . . . 6 ⊢ (((((𝜑 → ((𝜑 → 𝜓) → 𝜓)) → 𝜑) → (((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))) → ⊥)) → 𝜑) → (((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)))
3		merco1 1713	. . . . . 6 ⊢ ((((((𝜑 → ((𝜑 → 𝜓) → 𝜓)) → 𝜑) → (((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))) → ⊥)) → 𝜑) → (((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓))) → (((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))) → (((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))) → (𝜑 → ((𝜑 → 𝜓) → 𝜓)))))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))) → (((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))))
5		merco1lem9 1725	. . . . 5 ⊢ ((((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))) → (((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))) → (𝜑 → ((𝜑 → 𝜓) → 𝜓)))) → (((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))))
6	4, 5	ax-mp 5	. . . 4 ⊢ (((((𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) → ((𝜑 → 𝜓) → 𝜓)) → (𝜑 → ((𝜑 → 𝜓) → 𝜓))) → (𝜑 → ((𝜑 → 𝜓) → 𝜓)))
7	1, 6	ax-mp 5	. . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) → 𝜓))
8		merco1lem12 1728	. . 3 ⊢ ((𝜑 → ((𝜑 → 𝜓) → 𝜓)) → ((((𝜒 → 𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑 → 𝜓) → 𝜓)))
9	7, 8	ax-mp 5	. 2 ⊢ ((((𝜒 → 𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑 → 𝜓) → 𝜓))
10		merco1 1713	. 2 ⊢ (((((𝜒 → 𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑 → 𝜓) → 𝜓)) → ((((𝜑 → 𝜓) → 𝜓) → 𝜒) → (𝜑 → 𝜒)))
11	9, 10	ax-mp 5	1 ⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) → (𝜑 → 𝜒))

Syntax hints:   → wi 4  ⊥wfal 1552

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 207  df-tru 1543  df-fal 1553

This theorem is referenced by:  merco1lem15  1731  retbwax1  1735