Theorem merco1lem1 1713

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1712. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem1	⊢ (𝜑 → (⊥ → 𝜒))

Proof of Theorem merco1lem1

Step	Hyp	Ref	Expression
1		merco1 1712	. . . . 5 ⊢ (((((⊥ → 𝜑) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (((⊥ → 𝜑) → ⊥) → (𝜑 → ⊥)))
2		merco1 1712	. . . . 5 ⊢ ((((((⊥ → 𝜑) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (((⊥ → 𝜑) → ⊥) → (𝜑 → ⊥))) → (((((⊥ → 𝜑) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))))
3	1, 2	ax-mp 5	. . . 4 ⊢ (((((⊥ → 𝜑) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑)))
4		merco1 1712	. . . 4 ⊢ ((((((⊥ → 𝜑) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))) → (((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))))
5	3, 4	ax-mp 5	. . 3 ⊢ (((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑)))
6		merco1 1712	. . . . 5 ⊢ (((((⊥ → 𝜑) → (𝜑 → ⊥)) → ((𝜑 → (⊥ → 𝜑)) → ⊥)) → (𝜑 → (⊥ → 𝜑))) → (((𝜑 → (⊥ → 𝜑)) → ⊥) → (𝜑 → ⊥)))
7		merco1 1712	. . . . 5 ⊢ ((((((⊥ → 𝜑) → (𝜑 → ⊥)) → ((𝜑 → (⊥ → 𝜑)) → ⊥)) → (𝜑 → (⊥ → 𝜑))) → (((𝜑 → (⊥ → 𝜑)) → ⊥) → (𝜑 → ⊥))) → (((((𝜑 → (⊥ → 𝜑)) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑))))
8	6, 7	ax-mp 5	. . . 4 ⊢ (((((𝜑 → (⊥ → 𝜑)) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑)))
9		merco1 1712	. . . 4 ⊢ ((((((𝜑 → (⊥ → 𝜑)) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑))) → ((((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))) → (𝜑 → (𝜑 → (⊥ → 𝜑)))))
10	8, 9	ax-mp 5	. . 3 ⊢ ((((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))) → (𝜑 → (𝜑 → (⊥ → 𝜑))))
11	5, 10	ax-mp 5	. 2 ⊢ (𝜑 → (𝜑 → (⊥ → 𝜑)))
12		merco1 1712	. . . . 5 ⊢ (((((⊥ → 𝜑) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → (⊥ → 𝜒)) → (((⊥ → 𝜒) → ⊥) → (𝜑 → ⊥)))
13		merco1 1712	. . . . 5 ⊢ ((((((⊥ → 𝜑) → (𝜑 → ⊥)) → (𝜑 → ⊥)) → (⊥ → 𝜒)) → (((⊥ → 𝜒) → ⊥) → (𝜑 → ⊥))) → (((((⊥ → 𝜒) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))))
14	12, 13	ax-mp 5	. . . 4 ⊢ (((((⊥ → 𝜒) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑)))
15		merco1 1712	. . . 4 ⊢ ((((((⊥ → 𝜒) → ⊥) → (𝜑 → ⊥)) → (⊥ → 𝜑)) → (𝜑 → (⊥ → 𝜑))) → (((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒)) → (𝜑 → (⊥ → 𝜒))))
16	14, 15	ax-mp 5	. . 3 ⊢ (((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒)) → (𝜑 → (⊥ → 𝜒)))
17		merco1 1712	. . . . 5 ⊢ (((((⊥ → 𝜒) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)) → ((𝜑 → (⊥ → 𝜑)) → ⊥)) → (𝜑 → (⊥ → 𝜒))) → (((𝜑 → (⊥ → 𝜒)) → ⊥) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)))
18		merco1 1712	. . . . 5 ⊢ ((((((⊥ → 𝜒) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)) → ((𝜑 → (⊥ → 𝜑)) → ⊥)) → (𝜑 → (⊥ → 𝜒))) → (((𝜑 → (⊥ → 𝜒)) → ⊥) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥))) → (((((𝜑 → (⊥ → 𝜒)) → ⊥) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)) → (⊥ → 𝜒)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒))))
19	17, 18	ax-mp 5	. . . 4 ⊢ (((((𝜑 → (⊥ → 𝜒)) → ⊥) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)) → (⊥ → 𝜒)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒)))
20		merco1 1712	. . . 4 ⊢ ((((((𝜑 → (⊥ → 𝜒)) → ⊥) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → ⊥)) → (⊥ → 𝜒)) → ((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒))) → ((((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒)) → (𝜑 → (⊥ → 𝜒))) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → (𝜑 → (⊥ → 𝜒)))))
21	19, 20	ax-mp 5	. . 3 ⊢ ((((𝜑 → (⊥ → 𝜑)) → (⊥ → 𝜒)) → (𝜑 → (⊥ → 𝜒))) → ((𝜑 → (𝜑 → (⊥ → 𝜑))) → (𝜑 → (⊥ → 𝜒))))
22	16, 21	ax-mp 5	. 2 ⊢ ((𝜑 → (𝜑 → (⊥ → 𝜑))) → (𝜑 → (⊥ → 𝜒)))
23	11, 22	ax-mp 5	1 ⊢ (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 207  df-tru 1542  df-fal 1552

This theorem is referenced by:  retbwax4  1714  retbwax2  1715