Theorem merco1lem6 1718

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1710. (Contributed by Anthony Hart, 17-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem6	⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜒 → (𝜑 → 𝜓)))

Proof of Theorem merco1lem6

Step	Hyp	Ref	Expression
1		merco1lem5 1717	. . . . 5 ⊢ (((((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥) → ⊥) → ((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥))
2		merco1lem3 1715	. . . . 5 ⊢ ((((((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥) → ⊥) → ((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥)) → ((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → (((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥)))
3	1, 2	ax-mp 5	. . . 4 ⊢ ((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → (((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥))
4		merco1lem5 1717	. . . 4 ⊢ (((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → (((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥)) → ((𝜑 → 𝜓) → (((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥)))
5	3, 4	ax-mp 5	. . 3 ⊢ ((𝜑 → 𝜓) → (((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥))
6		merco1lem3 1715	. . 3 ⊢ (((𝜑 → 𝜓) → (((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → ⊥)) → (((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → 𝜑))
7	5, 6	ax-mp 5	. 2 ⊢ (((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → 𝜑)
8		merco1 1710	. 2 ⊢ ((((((𝜑 → 𝜓) → ⊥) → (𝜒 → ⊥)) → ⊥) → 𝜑) → ((𝜑 → (𝜑 → 𝜓)) → (𝜒 → (𝜑 → 𝜓))))
9	7, 8	ax-mp 5	1 ⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜒 → (𝜑 → 𝜓)))

Syntax hints:   → wi 4  ⊥wfal 1545

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 209  df-tru 1536  df-fal 1546

This theorem is referenced by:  merco1lem7  1719  merco1lem8  1721