Theorem merco1lem12 1727

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1712. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem12	⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜓))

Proof of Theorem merco1lem12

Step	Hyp	Ref	Expression
1		merco1lem3 1717	. . . . 5 ⊢ ((((𝜑 → 𝜏) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → ⊥)) → (𝜒 → ⊥)) → (𝜒 → (𝜑 → 𝜏)))
2		merco1 1712	. . . . 5 ⊢ (((((𝜑 → 𝜏) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → ⊥)) → (𝜒 → ⊥)) → (𝜒 → (𝜑 → 𝜏))) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜑)))
3	1, 2	ax-mp 5	. . . 4 ⊢ (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜑))
4		merco1lem9 1724	. . . 4 ⊢ ((((𝜒 → (𝜑 → 𝜏)) → 𝜑) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜑)) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜑))
5	3, 4	ax-mp 5	. . 3 ⊢ (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜑)
6		merco1lem11 1726	. . 3 ⊢ ((((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜑) → ((((𝜓 → 𝜑) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → ⊥)) → ⊥) → 𝜑))
7	5, 6	ax-mp 5	. 2 ⊢ ((((𝜓 → 𝜑) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → ⊥)) → ⊥) → 𝜑)
8		merco1 1712	. 2 ⊢ (((((𝜓 → 𝜑) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → ⊥)) → ⊥) → 𝜑) → ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 𝜓)))
9	7, 8	ax-mp 5	1 ⊢ ((𝜑 → 𝜓) → (((𝜒 → (𝜑 → 𝜏)) → 𝜑) → 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 207  df-tru 1542  df-fal 1552

This theorem is referenced by:  merco1lem13  1728  merco1lem14  1729