Theorem merco1lem9 1729

Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1717. (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 1728	. 2 ⊢ ((⊥ → 𝜑) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)))
2		merco1lem8 1728	. 2 ⊢ (((⊥ → 𝜑) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓))) → ((𝜑 → (𝜑 → 𝜓)) → (𝜑 → 𝜓)))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 → (𝜑 → 𝜓)) → (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 206  df-tru 1542  df-fal 1552

This theorem is referenced by:  merco1lem12  1732  merco1lem14  1734  merco1lem17  1737  merco1lem18  1738  retbwax1  1739