Theorem merco1lem9 1754

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1742. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
merco1lem9	⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))

Proof of Theorem merco1lem9

Step	Hyp	Ref	Expression
1		merco1lem8 1753	. 2 ⊢ ((⊥ → 𝜑) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)))
2		merco1lem8 1753	. 2 ⊢ (((⊥ → 𝜑) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4  ⊥wfal 1581

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-tru 1572  df-fal 1582

This theorem is used by:  merco1lem12  1757  merco1lem14  1759  merco1lem17  1762  merco1lem18  1763  retbwax1  1764