Theorem merco1lem3 1718

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1713. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem3	⊢ (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑))

Proof of Theorem merco1lem3

Step	Hyp	Ref	Expression
1		merco1lem2 1717	. . 3 ⊢ (((𝜑 → 𝜑) → ⊥) → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → ⊥))
2		retbwax2 1716	. . . 4 ⊢ ((((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑)) → (𝜑 → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑))))
3		merco1lem2 1717	. . . 4 ⊢ (((((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑)) → (𝜑 → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑)))) → ((((𝜑 → 𝜑) → ⊥) → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → ⊥)) → (𝜑 → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑)))))
4	2, 3	ax-mp 5	. . 3 ⊢ ((((𝜑 → 𝜑) → ⊥) → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → ⊥)) → (𝜑 → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑))))
5	1, 4	ax-mp 5	. 2 ⊢ (𝜑 → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑)))
6		merco1lem2 1717	. . 3 ⊢ (((𝜒 → 𝜑) → ⊥) → (((𝜑 → 𝜓) → (𝜒 → ⊥)) → ⊥))
7		retbwax2 1716	. . . 4 ⊢ ((((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑)) → ((𝜑 → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑))) → (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑))))
8		merco1lem2 1717	. . . 4 ⊢ (((((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑)) → ((𝜑 → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑))) → (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑)))) → ((((𝜒 → 𝜑) → ⊥) → (((𝜑 → 𝜓) → (𝜒 → ⊥)) → ⊥)) → ((𝜑 → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑))) → (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑)))))
9	7, 8	ax-mp 5	. . 3 ⊢ ((((𝜒 → 𝜑) → ⊥) → (((𝜑 → 𝜓) → (𝜒 → ⊥)) → ⊥)) → ((𝜑 → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑))) → (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑))))
10	6, 9	ax-mp 5	. 2 ⊢ ((𝜑 → (((𝜑 → 𝜑) → (𝜑 → ⊥)) → (𝜑 → 𝜑))) → (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑)))
11	5, 10	ax-mp 5	1 ⊢ (((𝜑 → 𝜓) → (𝜒 → ⊥)) → (𝜒 → 𝜑))

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:  merco1lem4  1719  merco1lem6  1721  merco1lem11  1727  merco1lem12  1728  merco1lem18  1734